Research Article Mean-Field Backward Stochastic Evolution …downloads.hindawi.com/journals/mpe/2014/718948.pdf · 2019-07-31 · 3. Mean-Field Backward Stochastic Evolution Equations

Research ArticleMean-Field Backward Stochastic Evolution Equations inHilbert Spaces and Optimal Control for BSPDEs

Ruimin Xu12 and Tingting Wu3

1 School of Mathematics Shandong University Jinan 250100 China2 School of Mathematics Shandong Polytechnic University Jinan 250353 China3 School of Mathematical Sciences Shandong Normal University Jinan 250014 China

Correspondence should be addressed to Ruimin Xu ruiminx126com

Received 31 March 2014 Revised 23 May 2014 Accepted 18 June 2014 Published 13 July 2014

Academic Editor Guangchen Wang

Copyright copy 2014 R Xu and T Wu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We obtain the existence and uniqueness result of the mild solutions to mean-field backward stochastic evolution equations (BSEEs)in Hilbert spaces under a weaker condition than the Lipschitz one As an intermediate step the existence and uniqueness result forthe mild solutions of mean-field BSEEs under Lipschitz condition is also established And then a maximum principle for optimalcontrol problems governed by backward stochastic partial differential equations (BSPDEs) of mean-field type is presented In thiscontrol system the control domain need not to be convex and the coefficients both in the state equation and in the cost functionaldepend on the law of the BSPDE as well as the state and the control Finally a linear-quadratic optimal control problem is given toexplain our theoretical results

1 Introduction

Backward stochastic evolution equations (BSEEs) in theirgeneral nonlinear form were introduced by Hu and Peng [1]in 1991 By the stochastic Fubini theorem and an extendedmartingale representation theoremHu andPeng [1] obtainedthe existence and uniqueness result of a so-called ldquomildsolutionrdquo under Lipschitz coefficients for semilinear BSEEsSince then BSEEs have been studied by a lot of authors andhave found various applications namely in the theory ofinfinite dimensional optimal control and the controllabilityfor stochastic partial differential equations (see eg [1ndash4] andthe papers cited therein) To relax the Lipschitz condition ofthe coefficients Mahmudov andMckibben [2] studied BSEEsunder a weaker condition than the Lipschitz one in Hilbertspaces Their approach extended the method proposed byMao [5] in which the author investigated BSDEs under aweaker condition which contains Lipschitz condition as aspecial case Our present work also investigates backwardstochastic evolution equations but with one main differ-ence to the setting chosen by the papers mentioned above

the coefficients of the BSEEs are allowed to depend on thelaw of the BSEEs

Recently mean-field approaches which can be usedto describe particle systems at the mesoscopic level haveattracted more and more researchersrsquo attention because oftheir great importance in applications For example mean-field approach can be used in statistical mechanics andphysics quantum mechanics and quantum chemistry eco-nomics finance game theory and optimal control theory(refer to [6ndash8] and the references therein) Mean-field BSDEswere deduced by Buckdahn et al [9] when they investi-gated a special mean-field problem in a purely stochasticapproach Buckdahn et al [7] studied the well posedness ofmean-field BSDEs and gave a probabilistic interpretation tosemilinear McKean-Vlasov partial differential equations Togive a probabilistic representation of the solutions for a classofMckean-Vlasov stochastic partial differential equations Xu[10] investigated the well-posedness of mean-field backwarddoubly stochastic differential equations with locally mono-tone coefficients

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 718948 18 pageshttpdxdoiorg1011552014718948

2 Mathematical Problems in Engineering

In this paper we investigate a new type of backwardstochastic evolution equations inHilbert spaces whichwe callmean-field BSEEs Mean-field implies that the coefficient ofthe BSEE depends on the law of the BSEE Specifically theBSEE we consider is defined as

119889119884 (119904) = minus 119860119884 (119904) 119889119904

minus E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

+ 119885 (119904) 119889119882 (119904)

119884 (119879) = 120585 119904 isin [0 119879]

(1)

in a Hilbert space 119867 where 119891 denotes a given measurablemapping 119879 is a fixed positive real number 119882(119904) is acylindrical Wiener process and 119860 represents the generatorof a strongly continuous semigroup 119890

119905119860 in 119867 with 119905 ge 0Precise interpretation of E1015840[119891(119904 1198841015840

(119904) 1198851015840(119904) 119884(119904) 119885(119904))] is

given in the following sections Based on the contractionmapping we firstly prove that (1) admits a unique mildsolution if the function 119891 is Lipschitz continuous Secondlyunder non-Lipschitz assumptions we obtain the existenceand uniqueness of the mild solution for mean-field BSEEby constructing a special Cauchy sequence The Lipschitzcondition is a special case of this non-Lipschitz condition (seeMao [5]) In addition we investigate the well-posedness ofmean-field stochastic evolution equations

We also study optimal control problems of stochasticsystems governed by mean-field BSPDEs in Hilbert spacesOur objective is to formulate a stochastic maximumprinciple(SMP) for the optimal control problem with an initial stateconstraint There is a vast literature on the theory of SMPAmong these papers Andersson and Djehiche [8] studiedthe optimal control problem for mean-field stochastic systemwhen the control domain is convex They obtained themaximum principle by a convex variational method By aspike variational technique Buckdahn et al [11] obtained ageneral maximum principle for a special mean-field stochas-tic differential equation (SDE) where the action space is notconvex Later Li [12] investigated the maximum principlefor more general SDEs of mean-field type with a convexcontrol domain Wang et al [13] were concerned with apartially observed optimal control problem of mean-fieldtype By using Girsanovrsquos theorem and convex variationthey derived the correspondingmaximum principle and gavean illustrative example to demonstrate the application ofthe obtained SMP Hafayed studied the mean-field SMP forsingular stochastic control in [14] and mean-field SMP forFBSDEs with Poisson jump processes in [15]

For the case of stochastic control systems in infinitedimensions on the assumption that the control domain isnot necessarily convex while the diffusion coefficient doesnot contain the control variable Hu and Peng [16] usedspike variation approach and Ekelandrsquos variational principleto establish the maximum principle for semilinear stochas-tic evolution control systems with a final state constraintMahmudov and Mckibben [2] obtained an SMP for stochas-tic control systems governed by BSEEs in Hilbert spacesRecently Fuhrman et al [17] deduced themaximumprinciple

for optimal control of stochastic PDEs when the controldomain is not necessarily convex

We establish necessary optimality conditions for thecontrol problem in the form of a maximum principle on theassumption that the control domain is not necessarily convexDue to the initial state constraint we first need to applyEkelandrsquos variational principle to convert the given controlproblem into a free initial state optimal control problemThenspike variation approach is used to deduce the SMP in themean-field framework In our control system not only thestate processes which are the unique mild solution of thegiven BSPDE but also the cost functional are of mean-fieldtype In other words they depend on the law of the BSPDEas well as the state and the control For this new controlledsystem the adjoint equation will turn out to be a mean-fieldstochastic evolution equation

Theplan of this paper is organized as follows In Section 2we introduce some notations which are needed in whatfollows In Section 3 the well-posedness of mean-field BSEE(1) is studied we first prove the existence and uniqueness of amild solution under the Lipschitz condition and investigatethe regular dependence of the solution (119884 119885) on (120585 119891)And then under the assumption that the coefficient is non-Lipschitz continuous a new result on the existence anduniqueness of the mild solution to (1) in Hilbert space isestablished which generalizes the result for the Lipschitzcase Section 4 is devoted to the regularity of mean-fieldstochastic evolution equations In Section 5 we derive thestochastic maximum principle for the BSPDE systems ofmean-field type with an initial state constraint and at thelast part of Section 5 an LQ example is given to show theapplication of our maximum principle An explicit optimalcontrol is obtained in this example

2 Preliminaries

The norm of an element 119909 in a Banach space 119865 is denotedby |119909|

119865or simply |119909| if no confusion is possible Γ 119867 and

119870 are three real and separable Hilbert spaces Scalar productis denoted by ⟨sdot sdot⟩ with a subscript to specify the space ifnecessaryL(Γ 119870) is the space of Hilbert-Schmidt operatorsfrom Γ to 119870 endowed with the Hilbert-Schmidt norm

Let (ΩFP) be a complete probability space A cylin-drical Wiener process 119882(119905) 119905 ge 0 in a Hilbert space Γ is afamily of linear mappings Γ rarr 119871

2(ΩFP) such that

(i) for every 119906 isin Γ 119882(119905)119906 119905 ge 0 is a real (continuous)Wiener process

(ii) for every 119906 V isin Γ and 119905 119904 ge 0 E(119882(119905)119906 sdot 119882(119904)V) =(119905 and 119904)⟨119906 V⟩

Γ

By F119905 119905 isin [0 119879] we denote the natural filtration of 119882

augmented with the familyN of P-null sets ofF119879

F119905= 120590 (119882 (119904) 119904 isin [0 119905]) orN (2)

The filtration (F119905)119905ge0

satisfies the usual conditions Allthe concepts of measurability for stochastic processes (egadapted etc) refer to this filtration

Mathematical Problems in Engineering 3

Next we define several classes of stochastic processes withvalues in a Hilbert space119867

(I) H2

F ([0 119879]119867) denotes the set of (classes of 119889P times

119889119905 ae equal) measurable random processes 120595119905 119905 isin

[0 119879] which satisfy

(i) Eint1198790|120595

119905|2119889119905 lt +infin

(ii) 120595119905isF

119905measurable for ae 0 le 119905 le 119879

Evidently H2

F (0 119879119867) is a Banach space en-dowed with the canonical norm

10038171003817100381710038171205951003817100381710038171003817 = Eint

119879

0

100381610038161003816100381612059511990410038161003816100381610038162

119889119904

12

(3)

(II) S2

F ([0 119879]119867) denotes the set of continuous randomprocesses 120595

119905 119905 isin [0 119879] which satisfy

(i) E(sup0le119905le119879

|120595119905|2) lt +infin

(ii) 120595119905isF


(III) 1198710(ΩFP 119867) denotes the space of all 119867 valued F-measurable random variables

(IV) For 1 le 119901 lt infin 119871119901(ΩFP 119867) is the space of allF-measurable random variables such that E[|120585|119901] lt infin

(V) For any 120573 isin R introduce the norm

1003817100381710038171003817(119910 119911)10038171003817100381710038172

120573119905= Eint

119879

119905

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904 (4)

on the Banach space

K120573[119905 119879] = S

2

F ([119905 119879] 119867) timesH2

F ([119905 119879] L (Γ119867)) (5)

For 0 lt 119879 lt infin all the norms sdot 120573119905

with different 120573 isin R areequivalentK[0 119879] = K

0[0 119879] is the Banach space endowed

with the norm

1003817100381710038171003817(119910 119911)10038171003817100381710038172

= Eint119879

0

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904 (6)

The following result on BSEEs (see Lemma 2 in Mahmu-dov and McKibben [2]) will play a key role in proving thewell-posedness of mean-field BSEEs

Lemma 1 Let 119867 be a Hilbert space and let 119860 119863(119860) sub

119867 rarr 119867 be a linear operator which generates a 1198620-semigroup

119878(119905) 0 le 119905 le 119879 on 119867 For any (120585 119891) isin 1198712(ΩF

119879P 119867) times

H2

F ([0 119879]119867) the following equation

119884 (119905) = 119878 (119879 minus 119905) 120585 + int

119879

119905

119878 (119904 minus 119905) 119891 (119904) 119889119904

+ int

119879

119905

119878 (119904 minus 119905) 119885 (119904) 119889119882 (119904) 119875-119886119904(7)

has a unique solution inK120573[0 119879] moreover

E sup119905le119904le119879

1198902120573119904

|119884 (119904)|2+ Eint

119879

119905

1198902120573119904

|119885 (119904)|2119889119904

le 241198722

119878(119890

2120573119879E100381610038161003816100381612058510038161003816100381610038162

+1

2120573int

119879

119905

1198902120573119903

E1003816100381610038161003816119891 (119903)

10038161003816100381610038162

119889119903)

(8)

where119872119878= sup119878(119905)B(119867) 0 le 119905 le 119879 andB(119867) is the space

of bounded linear operators on119867

3 Mean-Field Backward StochasticEvolution Equations

In this section we study the existence and uniqueness resultof mild solutions to mean-field BSEEs in a Hilbert space 119867To this end we firstly recall some notations introduced byBuckdahn et al [7]

Let (ΩFP) = (Ω times ΩF otimes FP otimes P) be the(noncompleted) product of (ΩFP) with itself and wedefine F = F

119905= F otimes F

119905 0 le 119905 le 119879 on this product

space A random variable 120585 isin 1198710(ΩFP 119867) originally

defined on Ω is extended canonically to Ω 1205851015840(120596

1015840 120596) =

120585(1205961015840) (120596

1015840 120596) isin Ω = Ω times Ω For any 120578 isin 119871

1(ΩFP) the

variable 120578(sdot 120596) Ω rarr 119870 belongs to 1198711(ΩFP)P(119889120596) aswhose expectation is denoted by

E1015840[120578 (sdot 120596)] = int

Ω

120578 (1205961015840 120596)P (119889120596

1015840) (9)

Note that E1015840[120578] = E1015840[120578(sdot 120596)] isin 1198711(ΩFP) and

E [120578] (= intΩ

120578119889P = intΩ

E1015840[120578 (sdot 120596)]P (119889120596)) = E [E

1015840[120578]]

(10)

Themean-field BSEE we consider has the following formfor any given measurable mapping 119891 [0 119879] times119867timesL(Γ119867)times

119867 timesL(Γ119867) rarr 119867 and 120585 isin 1198712(ΩF119879P 119867)

119889119884 (119904) = minus 119860119884 (119904) 119889119904

minus E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

+ 119885 (119904) 119889119882 (119904)

119884 (119879) = 120585 119904 isin [0 119879]

(11)

where 119860 119863(119860) sub 119867 rarr 119867 is the generator of a stronglycontinuous semigroup 119890119905119860 119905 ge 0 in the Hilbert space119867 withthe notation119872

119860≜ sup

119905isin[0119879]|119890119905119860|


Definition 2 We say that a pair of adapted processes (119884 119885)is a mild solution of mean-field BSEE (11) if (119884 119885) isin

S2

F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867)) and for all 119905 isin [0 119879]

119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

119890119860(119904minus119905)

E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(12)

Remark 3 We emphasize that the coefficient of (11) can beinterpreted as

E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] (120596)

= E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (120596 119904) 119885 (120596 119904))]

= intΩ

119891 (1205961015840 120596 119904 119884 (120596

1015840 119904) 119885 (120596

1015840 119904) 119884 (120596 119904) 119885 (120596 119904))

times P (1198891205961015840)

(13)

31 Lipschitz Case Now we study the existence and unique-ness ofmild solutions tomean-field BSEE (11) under Lipschitzconditions For119891 [0 119879] times 119867 times L(Γ119867) times 119867 times L(Γ119867) rarr

119867 assume the following(A1) There exists an 119871 gt 0 such that

10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 119871 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

+100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(14)

for all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 119911

1015840

119894 119911

119894isin L(Γ119867) (119894 =

1 2)(A2) 119891(sdot 0 0 0 0) isin H2

F ([0 119879]119867)We have the following theorem

Theorem 4 For any random variable 120585 isin 1198712(ΩF

119879P 119867)

under (A1) and (A2) mean-field BSEE (11) admits a uniquemild solution (119884 119885) isin S2

F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867))

Proof Consider the following

Step 1 For any (119910 119911) isin S2

F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867))BSEE119884 (119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904) 0 le 119905 le 119879

(15)

has a unique solution In order to get this conclusion wedefine

119891(119910119911)

(119904 120583 ]) = E1015840[119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 120583 ])] (16)

Then (15) can be rewritten as

119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

119890119860(119904minus119905)

119891(119910119911)

(119884 (119904) 119885 (119904)) 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(17)

Due to (A1) for all (1205831 ]

1) (120583

2 ]

2) isin 119867timesL(Γ119867)119891 satisfies

10038161003816100381610038161003816119891(119910119911)

(1205831 ]

1) minus 119891

(119910119911)(120583

2 ]

2)10038161003816100381610038161003816

2

le 119871 (10038161003816100381610038161205831 minus 1205832

10038161003816100381610038162

+1003816100381610038161003816]1 minus ]

2

10038161003816100381610038162

)

(18)

According to Theorem 31 in [1] BSEE (15) has a uniquesolution

Step 2 FromStep 1 we can define amappingΦ (119884(sdot) 119885(sdot)) =

Φ[(1199101015840(sdot) 119911

1015840(sdot))] K[0 119879] rarr K[0 119879] through

119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904) 0 le 119905 le 119879

(19)

For any (119910119894 119911119894) isin K[0 119879] we set (119884119894 119885

119894) = Φ[(119910

119894 119911

119894)] 119894 =

1 2 (119910 119911) = (1199101minus119910

2 119911

1minus119911

2) and (119884 119885) = (119884

1minus119884

2 119885

1minus119885

2)

Then from Lemma 1 we have

E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E


times [int

119879

0

1198902120573119904

100381610038161003816100381610038161003816E1015840[119891 (119904 (119910

1

(119904))1015840

(1199111

(119904))1015840

1198841

(119904) 1198851

(119904))

minus 119891 (119904 (1199102

(119904))1015840

(1199112

(119904))1015840

1198842

(119904) 1198852

(119904))]100381610038161003816100381610038161003816

2

119889119904]

le12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(E [1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2]

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(20)

If we set 120573 = 361198722

1198601198712max119879 1 then

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le 119879 sdot E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

1198601198712max 119879 1120573

E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=1

3E [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(21)

That is

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le1

2Eint

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904

(22)

The estimate (22) shows that Φ is a contraction on thespaceK

120573[0 119879] with the norm

(119884 119885)2

120573= Eint

119879

0

1198902120573119904

(|119884 (119904)|2+ |119885 (119904)|

2) 119889119904 (23)

With the contraction mapping theorem there admits aunique fixed point (119884 119885) isin K

120573[0 119879] such that Φ(119884 119885) =

(119884 119885) On the other hand from Step 1 we know thatif Φ(119884 119885) = (119884 119885) then (119884 119885) isin S2

F ([0 119879]119867) times

H2

F ([0 119879]L(Γ119867)) which is the unique mild solution of(11)

Arguing as the previous proof we arrive at the followingassertion in a straightforward way

Corollary 5 Suppose that for all 120572 in a metric space 119865 119891120572is

a given function satisfying (A1) and (A2) with 119871 independenton 120572 Also suppose that

E1015840[119891

120572(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

997888rarr E1015840[119891

1205720(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

(24)

in 1198712([0 119879]119867) as 120572 rarr 1205720for all (119884 119885) isin S2

F ([0 119879]119867) times

H2

F ([0 119879]L(Γ119867))If we denote by (119884(120585 120572) 119885(120585 120572)) the mild solution of (11)

corresponding to the functions 119891120572and to the final data 120585 isin

1198712(ΩF

119879P 119867) then the map (120572 120585) rarr (119884(120585 120572) 119885(120585 120572))

is continuous from 119865 times 1198712(ΩF

119879P 119867) to S2

F ([0 119879]119867) times

H2

F ([0 119879]L(Γ119867))

32 Non-Lipschitz Case This subsection is devoted to findingsome weaker conditions than the Lipschitz one under whichthe mean-field BSEE has a unique solution To state our mainresult in this section we suppose the following

(A3) For all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 1199111015840

119894 119911

119894isin L(Γ119867) (119894 =

1 2) there exists an 119897 gt 0 such that

10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 120579 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

) + 120579 (10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

)

+ 119897 (100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(25)

where 120579 R+rarr R+ is a concave increasing function such

that 120579(0) = 0 120579(119906) gt 0 for 119906 gt 0 and int0+(119889119906120579(119906)) = infin

InMao [5] the author gave three examples of the function120579(sdot) to show the generality of condition (A3) From theseexamples we can see that Lipschitz condition (A1) is a specialcase of the given condition (A3)

Since 120579 is concave and 120579(0) = 0 there exists a pair ofpositive constants 119886 and 119887 such that

120579 (119906) le 119886 + 119887119906 (26)

for all 119906 ge 0 Therefore under assumptions (A2) and(A3) 119891(sdot 1199101015840(sdot) 1199111015840(sdot) 119910(sdot) 119911(sdot)) isin H2

F ([0 119879]119867) whenever


1199101015840(sdot) 119910(sdot) isin S2

F ([0 119879]119867) and 1199111015840(sdot) 119911(sdot) isin H2

F ([0 119879]L(Γ119867))

By Picard-type iteration we now construct an approxi-mate sequence using which we obtain the desired result Let1198840(119905) equiv 0 and for 119899 isin N let 119884

119899 119885

119899 be a sequence in

S2

F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867)) defined recursively by

119884119899(119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)

times119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885119899(119904) 119889119882 (119904)

(27)

on 0 le 119905 le 119879 From Theorem 4 (27) has a unique mildsolution (119884

119899(119905) 119885

119899(119905))

In order to give the main result we need to prepare thefollowing lemmas about the properties of (119884

119899(119905) 119885

119899(119905)) 119905 isin

[0 119879]

Lemma 6 Under hypotheses (A2) and (A3) there existpositive constants 119862

1and 119862

2such that

(i)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905)

(ii)Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904 le 21198621exp (119879 minus 119905)

(iii)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le 1198622int

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(28)

for all 119905 isin [0 119879] and 119899 ge 1

Proof Using the hypotheses (A2) and (A3)with 120579(119906) le 119886 + 119887119906

yields

10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

le 2120579 (100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 2120579 (1003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

)

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

le 4119886 + 2119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 21198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

(29)

Then it follows from Lemma 1 that

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) + Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))]

10038161003816100381610038161003816

2

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904 10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le 1198621+24119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904[119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 119897100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

]] 119889119904

= 1198621+48119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

(30)

where

1198621= 24119872

2

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[2119886 +1003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

] 119889119904 + 1

(31)

If we set 120573 = 961198722

119860max119887 119897 we can obtain

sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) +1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

(32)

An application of the Gronwall inequality now implies

sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905

2)

le 21198621exp (119879 minus 119905)

(33)

Point (i) of Lemma 6 is now proved


From formula (32) we know that

int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 21198621+ int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

le 21198621+ 2119862

1int

119879

119905

exp (119879 minus 119904) 119889119904

= 21198621exp (119879 minus 119905)

(34)

This proves point (ii) of the LemmaTo prove point (iii) we note that

E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

]

le E1015840[120579 (

100381610038161003816100381610038161198841015840

119899(119904) minus 119884

1015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 119897100381610038161003816100381610038161198851015840

119899+1(119904) minus 119885

1015840

119899(119904)10038161003816100381610038161003816

2

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

= E [120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

(35)

By Lemma 1 we have

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

+ Eint119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))]

10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904

E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)

+ 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

] 119889119904

(36)

We can choose 120573 gt 0 sufficiently large such that

(1 minus24119872

2

119860119897

120573)Eint

119879

119905

11989021205731199041003817100381710038171003817119885119899+1

(119904) minus 119885119899(119904)10038171003817100381710038172

119889119904 ge 0 (37)

Then

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(38)

where we set 1198622= (24119872

2

119860120573)119890

2120573119879

We divide the interval [0 119879] into subintervals 0 = 1205910lt

1205911lt sdot sdot sdot lt 120591

119898= 119879 by setting 120591

119896= 119896120575 119896 = 1 2 3 119898 with

120575 = 119879119898

Lemma 7 For all 119905 isin [120591119896minus1

120591119896] define

1198623= 119862

2120579 (2119862

1exp (119879))

1205931198961(119905) = 119862

3(120591119896minus 119905)

120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 119899 ge 1

(39)

Then for all 119899 ge 1 the following inequality holds for a suitable120575 gt 0

0 le 120593119896119899(119905) le 120593

119896119899minus1(119905) le sdot sdot sdot le 120593

1198961(119905) (40)

Proof Firstly it needs to be verified that for all 119905 isin [120591119896minus1

120591119896]

the following inequality

1205931198962(119905) = 119862

2int

120591119896

119905

120579 (1205931198961(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (1198623(120591119896minus 119904)) 119889119904

le 1198623(120591119896minus 119905) = 120593

1198961(119905)

(41)

holds provided 120575 gt 0 is chosen sufficiently smallActually this inequality holds provided that

1198622120579 (119862

3(120591119896minus 119905)) le 119862

3= 119862

2120579 (2119862

1exp (119879)) (42)


or

1198623(120591119896minus 119905) = 119862

2120579 (2119862

1exp (119879)) (120591

119896minus 119905) le 2119862

1exp (119879)

(43)

Since 1198621gt 1 from 120579(119906) le 119886 + 119887119906 the above inequality holds

if

1198622(119886 + 119887) (120591

119896minus 119905) le 1 (44)

Thus (41) holds for any 119905 isin [120591119896minus1

120591119896] 119896 = 1 2 119898 if 120591

119896minus

120591119896minus1

le 11198622(119886 + 119887) Therefore we can choose a sufficiently

large119898 isin N such that 120575 = 119879119898 le 11198622(119886 + 119887) Clearly such a

120575 only depends on 119886 119887 119897 119879 and119872119860

Now assume that (40) holds for some 119899 ge 2 Then wehave

120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 le 119862

2int

120591119896

119905

120579 (120593119896119899minus1

(119904)) 119889119904

= 120593119896119899(119905) forall119905 isin [120591

119896minus1 120591

119896]

(45)

This completes the proof

Now we can give the main result of this section

Theorem 8 Assume that (A2) and (A3) hold Then thereexists a unique mild solution (119884 119885) to (11)


Uniqueness To show the uniqueness let both (119884 119885) and( 119885) be solutions of (11) For any 120573 gt 0 similar to the proofof (36) one can obtain

E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) + Eint119879

119905

119890212057311990410038161003816100381610038161003816

119885 (119904) minus 119885 (119904)10038161003816100381610038161003816

2

119889119904

le24119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

) + 11989710038161003816100381610038161003816119885 (119904) minus 119885 (119904)

10038161003816100381610038161003816

2

] 119889119904

(46)

That is if 120573 is sufficiently large

E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

)] 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

119890212057311990310038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)119889119904

(47)

An application of Bihari inequality yields

E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) = 0 (48)

So 119884(119905) = (119905) for all 119905 isin [0 119879] as It then follows from (46)that 119885(119905) = 119885(119905) for all 119905 isin [0 119879] as as well This establishesthe uniqueness

ExistenceWe claim that the sequence (119884119899 119885

119899) defined by (27)

satisfies

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

997888rarr 0 forall0 le 119905 le 119879 (49)

as 119899 rarr infinIndeed for all 119905 isin [120591

119896minus1 120591

119896] we set 120593

119896119899(119905) =

E sup119904isin[119905120591119896]

1198902120573119904|119884119899+1

(119904) minus 119884119899(119904)|

2 By Lemmas 6 and 7

1205931198961(119905) = E sup

119904isin[119905120591119896]

119890212057311990410038161003816100381610038161198842 (119904) minus 1198841 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119904le119903le120591119896

119890212057311990310038161003816100381610038161198841 (119903) minus 1198840 (119903)

10038161003816100381610038162

)119889119904

le 1198622int

120591119896

119905

120579 (21198621exp (120591

119896minus 119905)) 119889119904

le 1198622120579 (2119862

1exp (119879)) (120591

119896minus 119905) = 119862

3(120591119896minus 119905) = 120593

1198961(119905)

(50)

Suppose that 120593119896119899(119905) le 120593

119896119899(119905) holds for some 119899 ge 1

According to Lemma 6(iii) and Lemma 7 for all 119905 isin [120591119896minus1

120591119896]

we obtain

120593119896119899+1

(119905) = E sup119904isin[119905120591119896]

11989021205731199041003816100381610038161003816119884119899+2 (119904) minus 119884119899+1 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119903isin[119904120591119896]

11989021205731199031003816100381610038161003816119884119899+1 (119903) minus 119884119899 (119903)

10038161003816100381610038162

)119889119904

= 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904

le 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 120593

119896119899+1(119905)

(51)

This implies that for all 119899 isin N

120593119896119899(119905) le 120593

119896119899(119905) (52)

By definition 120593119896119899(sdot) is continuous on [120591

119896minus1 120591

119896] Note that

for each 119899 ge 1 120593119896119899(sdot) is decreasing on [120591

119896minus1 120591

119896] and for each

119905 119896 120593119896119899(119905) is a nonincreasing sequence Therefore we define

the function 120593119896(119905) by 120593

119896119899(119905) darr 120593

119896(119905) It is easy to verify that

120593119896(119905) is continuous and nonincreasing on [120591

119896minus1 120591

119896] By the

definitions of 120593119896119899(119905) and 120593

119896(119905) we get

120593119896(119905) = lim

119899rarrinfin

1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (120593119896(119904)) 119889119904

(53)

for all 120591119896minus1

le 119905 le 120591119896 Since int

0+(119889119906120579(119906)) = infin the Bihari

inequality implies

120593119896(119905) = 0 for each 119905 isin [120591

119896minus1 120591

119896] (54)

For each 119896 isin 1 2 119898 (52) and (54) yield

lim119899rarrinfin

120593119896119899(119905) = 0 (55)


Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)

as 119899 rarr infin and this proves the assertion (49)By (36) we obtain

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)

Applying (49) to the above formula we see that (119884119899 119885

119899) is

a Cauchy (hence convergent) sequence in S2

F ([0 119879]119867) times

H2

F ([0 119879]L(Γ119867)) denote the limit by (119884 119885) Now letting119899 rarr infin in (27) we obtain that

119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)

holds on the entire interval [0 119879]The theorem is nowproved

To illustrate the application of the obtained existenceand uniqueness result we consider the example of backwardstochastic partial differential equations (BSPDEs) of mean-field type

Example 9 Let O be an open bounded domain in R119899

with uniformly 1198622 boundary 120597O let 119861(119905) be a standard 119899-dimensional Brownian motion (equipped with the normalfiltration) and let 120585 O rarr R be an F

119879-measurable

random variable We also let 119871 denote the semielliptic partialdifferential operator on 1198622

(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)

The aim is to study the solvability of the following initialboundary value problem

119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))

119892 [0 119879] times O timesR timesL (R119899 119871

2

(O))

timesR timesL (R119899 119871

2

(O)) 997888rarr R

(61)

The following assumptions will have to be in force

(H1) 119886119894119895 119887

119894 R119899

rarr R are uniformly continuous andbounded and satisfy the usual uniform ellipticitycondition sum119899

119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|

2 for some 120582 gt 0

and all 119909 isin O 119908 isin R119899(H2) 119892 is measurable in (119905 119909 119910 119910 119911) and continuous in

( 119911) and there exists 119862 gt 0 such that1003816100381610038161003816119892 (119905 119909 1199101 1 1199101 1199111) minus 119892 (119905 119909 1199102 2 1199102 1199112)

1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)

for all 0 le 119905 le 119879 119909 isin O 1199101 119910

2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))

Then we are now in a position of showing existence anduniqueness of the solution of BSPDEs (60)

Theorem 10 If (H1) and (H2) are satisfied then the mean-field BSPDE (60) has a unique mild solution (119884 119885) isin

1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))

Proof Let119867 = 1198712(O) and119870 = R119899 Define the operator 119860 by

119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)

It is shown in [17] (see Example 21 in [17]) that 119860 generatesa strongly continuous semigroup on 119867 Define the maps 119891

[0 119879] times 119867 timesL(119870119867) times 119867 timesL(119870119867) rarr 119867 by

119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)


for all 0 le 119905 le 119879 119909 isin O With these identifications(60) can be written in the form of (11) By (H2) we know119891 satisfy condition (A1) Hence an application of Theorem 4concludes that (60) has a unique mild solution (119884 119885) isin

1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))

4 Mean-Field Stochastic Evolution Equations

Let 119882(119905) 119905 isin [0 119879] be a cylindrical Wiener process withvalues in a Hilbert space Γ defined on a probability space(ΩFP) We fix an interval [119905 119879] sub [0 119879] and considerthe stochastic evolution equations of mean-field type for anunknown process 119883(119904) 119904 isin [119905 119879] with values in a Hilbertspace 119870

119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)

where operator 119861 is the generator of a strongly continuoussemigroup 119890

119905119861 119905 ge 0 in the Hilbert space 119870 with 119872119861≜

sup119905isin[0119879]

|119890119905119861|

By a mild solution of (65) we mean an F119904-measurable

process 119883(119904) 119904 isin [119905 119879] with continuous paths in 119870 suchthat P-as

119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)

We suppose the following(A4) 119887 [0 119879] times 119870 times 119870 rarr 119870 is a measurable mapping

which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)

for some constant 1198711gt 0

(A5) The mapping 120590 [0 119879] times 119870 times 119870 rarr L(Γ 119870) fulfillsthat for every V isin Γ the map 120590V [0 119879] times 119870 times 119870 rarr 119870

is measurable for every 119904 gt 0 119905 isin [0 119879] 1199091015840 1199101015840 119909 119910 isin 119870119890119904119861120590(119905 119909

1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)

for some constants 1198712gt 0 and 120574 isin [0 12)

Theorem 11 Under assumptions (A3) and (A4) (65) has aunique mild solution119883(sdot) isin S2

F ([119905 119879] 119870)

The proof is constructed in two steps like that ofTheorem 4 and it uses standard arguments for stochastic evo-lution equations introduced in the proof of Proposition 32 in[3] Since the proof is straightforward we prefer to omit it

Remark 12 In our paper Lipchitz condtion (A4) is givento get the well-posedness of mean-field stochastic evolutionequations In fact (A4) can be replaced by a weaker conditionsuch as (A3) We just give the condition (A4) for simplicity

From standard arguments we can also get the followingcontinuous dependence theorem

Corollary 13 Assume that for all 120572 in a metric space 119865(119887120572 120590

120572) satisfy (A4) and (A5)with119871

1and119871

2independent of 120572

Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)

as 120572 rarr 1205720for all119883 isin S2

F ([0 119879] 119870)If we denote by 119883120572

(sdot) the mild solution of mean-field SEE(65) corresponding to the functions (119887

120572 120590

120572) and to the initial

data 119909 then we have

sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)

5 Maximum Principle for BSPDEs ofMean-Field Type

51 Formulation of the Problem Let O isin R119899 be a boundedopen set with smooth boundary 120597O and let 119880 the space ofcontrols be a separable real Hilbert space We denote

U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω

997888rarr 119880 is F otimesF119905-progressively measurable

(71)


An element ofU is called an admissible controlFor any V isin U we consider the following controlled

BSPDE system in the state space119867 = 1198712(O) (norm | sdot | scalar

product ⟨sdot sdot⟩)

119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)

where 119860 is a partial differential operator 119891 [0 119879] timesO times119867 times

L(Γ119867)times119867timesL(Γ119867)times119880 rarr 119867 and 120585 isin 1198712(ΩF119879P 119867)

The cost functional is given by

119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where

ℎ [0 119879] times O times 119867 timesL (Γ119867) times 119867 timesL (Γ119867) times 119880 997888rarr R

119892 O times 119867 times119867 997888rarr R

(74)

Our purpose is to minimize the functional 119869(sdot) over Uadsubject to the following state constraint

EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where

Φ O times 119867 times119867 997888rarr R (76)

An admissible control 119906 isin Uad that satisfies

119869 (119906) = minVisinUad

119869 (V) (77)

is called optimalThrough what follows the following assumptions will be

in force

(L1) 119860 is a partial differential operator with appropri-ate boundary conditions We assume that 119860 is theinfinitesimal generator of a strongly continuous semi-group 119890119905119860 119905 ge 0 in119867 Moreover for every 119905 isin [0 119879]119890

119905119860119891

1198712(O) le 119872

119860119891

1198712(O) for some constant 119872

119860

independent of 119905 and 119891(L2) 119891 ℎ 119892 and Φ are continuously Gateaux differen-

tiable with respect to (1199101015840 1199111015840 119910 119911) 119891 is continuouslyGateaux differentiable with respect to V and ℎ iscontinuous with respect to V

(L3) The derivatives of 119891 ℎ 119892 and Φ are Lipschitzcontinuous and bounded by

100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)

where 119862 is a positive constant

Obviously according to Theorem 4 state equation (72)has a unique mild solution under the above assumptions

Remark 14 We can define the second order differentialoperator

(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)

By Example 9 119860 fulfills assumption (L1) if 119886119894119895 119887

119894satisfy

condition (H1)

52 Variation of the Trajectory Let 119906 be an optimal controlwith (119884(sdot) 119885(sdot)) being the corresponding optimal state Let120576 gt 0 and [119903 119903 + 120576] sube [0 119879] For any given V isin Uad weintroduce the spike variation of the control 119906(sdot)

119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)

It is clear that 119906120576(sdot) isin UadLet (119884120576

(sdot) 119885120576(sdot)) be the trajectory corresponding to 119906120576(sdot)

We use the following short notation for brevity

119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)

Consider the following equation

119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)

Since the coefficients in (82) are bounded it is easy to checkthat there exists a unique mild solution such that

E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)

We have the following estimate


Theorem 15 There holds

lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)

For simplicity let us define

Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)

By the definition of (119884120576

119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)

is the mild solution of

119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)

For any 120573 gt 0 according to Lemma 1 we obtain

E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)

By condition (L3) we have

E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)

Combined with (91) (90) yields

E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2

]119889119904 for example

Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)

The inequality above holds due to the boundedness of|119870

120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2

119889120582 Indeed Assumption (L3) im-plies the boundedness of 119891

119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is

the solution of mean-field BSEE (82) It can be easy to check119870120576

119904is bounded since the coefficients in (82) are boundedOn the other hand

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576

119904) is the mild solution of the following equation

119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885

119904) is the mild solution of

119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)

By the definition of 119906120576119905 according to (L2) we have

E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)

in 1198712([0 119879]119867) as 120576 rarr 0 Using the continuous dependencetheorem Corollary 5 we obtain

(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)


Combining (98) with (104) we finally haveEint

119879

119905E1015840[|1198682

119904|2

]119889119904 rarr 0 as 120576 rarr 0The required result (93) follows by using the similar

estimations for 1198681119904 1198683

119904 and 1198684

119904

Note that 1 minus 1921198722

1198601198622120573 gt 0 if 120573 is sufficiently large

Now to prove the desired result (84) it suffices to applyGronwallrsquos lemma and estimate (93) to inequality (92)

To deal with the state constraint (75) we need to recall theEkeland variational principle

Lemma 16 (Ekelandrsquos variational principle see [16 Lemma41]) Let (119878 119889) be a complete metric space and let 119865(sdot) 119878 rarr

R be lower semicontinuous and bounded from below If for 120588 gt0 there exists 119906 isin 119878 such that

119865 (119906) le infVisin119878

119865 (V) + 120588 (105)

then there exists 119906120588 isin 119878 satisfying

(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588

(iii) 119865 (119906120588) le 119865 (V) + 120588 sdot 119889 (119906120588 119906) forallV = 119906120588

(106)

Now fix V isin Uad and set

119878 = V (sdot) isin Uad | sup0le119905le119879

E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2

119889 (V (sdot) V (sdot)) = 119898 119905 isin [0 119879] | E1003816100381610038161003816V119905 minus V

119905

10038161003816100381610038162

gt 0

forallV (sdot) V (sdot) isin 119878

(107)

where119898 denotes the Lebesgue measure on RThe following result is proved as Proposition 41 in [16]

Lemma 17 (119878 119889(sdot sdot)) is a complete metric space and 119869120588 is

continuous and bounded on 119878 where

119869120588

(V (sdot)) = (119869 (V (sdot)) minus 119869 (119906 (sdot)) + 120588)2

+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12

forallV (sdot) isin 119878(108)

and (119884 119885) is the mild solution of (72) corresponding to thecontrol V

Now we consider the following free initial state optimalcontrol problem

infV(sdot)isin119878

119869120588

(V (sdot)) (109)

It is easy to check that

0 le infV(sdot)isin119878

119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)

According to Ekelandrsquos variational principle there exists a119906120588(sdot) isin 119881 such that

(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588

(V (sdot)) + 120588119889 (119906120588 (sdot) 119906 (sdot)) forallV (sdot) isin 119878(111)

Using the spike variationmethod we can construct 119906120576120588(sdot) isin 119878as follows

119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)

It is clear that 119889(119906120576120588(sdot) 119906120588(sdot)) le 120576 Let (119884120576120588(sdot) 119885

120576120588(sdot)) (resp

(119884120588(sdot) 119885

120588(sdot))) be the solution of (72) with respect to the

control 119906120576120588(sdot) (resp 119906120588(sdot)) Following (82) (119870120576120588

119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)

ByTheorem 15 we know that

lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)

The proof of the following proposition is technical butbased on the arguments above and we omit it

Proposition 18 One has

1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)

53 Variational Inequality and Adjoint Equation In thissubsection the adjoint process is introduced to deduce thevariational inequality

If we set V(sdot) = 119906120576120588(sdot) in (111) and notice that

119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)

as 120576 rarr 0 according to (115)As |119897120588

1|2

+ |119897120588

2|2

= 1 for all 120588 gt 0 we know that there existsa subsequence of 119897120588

1 119897120588

2 (still denoted by 119897120588

1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)

Combining (115) (117) with (118) we get

minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)

Next we introduce the adjoint equation corresponding tovariational equation (113) whose solution is denoted by119875120588(119905)

119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)

where 119860lowast is the 1198712(O)-adjoint operator of 119860 Under assump-tions (L1)ndash(L3) this is a linear mean-field SEE with boundedcoefficients An application ofTheorem 11 implies that it has aunique adaptedmild solution such that119875120588(119905) isin S2

F ([0 119879] 119870)When 120588 rarr 0 according to Corollaries 5 and 13 119875120588(119905)

converges to 119875(119905) where

119875 (119905) isin S2

F ([0 119879] 119870) (124)

is the solution of the following equation

119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)

The following proposition which formally follows fromProposition 18 gives the relation between 119875120588(119905) and119870120576120588

119905

Proposition 19 Consider the following

E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)

The following theorem constitutes the main contributionof this section themaximumprinciple for the BSPDE controlsystem

Theorem 20 Let assumptions (L1)ndash(L3) hold Suppose 119906(sdot) isan optimal control and (119884(sdot) 119885(sdot)) is the corresponding optimalstate trajectory for the BSPDE control systems (72) and (73)with the initial state constraint (75) Then there exists 119875(119905) isinS2

F ([0 119879] 119870) which satisfies (125) such that

H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)

whereH [0 119879]times119867timesL(Γ119867)times119867timesL(Γ119867)times119880times119870 rarr R

is the Hamiltonian function defined by

H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)

Proof By (122) and Proposition 19 we obtain

minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)

Letting 120576 rarr 0+ in (129) we derive for ae 120591 isin [0 119879]

minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)

for all V isin UadFinally taking 120588 rarr 0 in (130) we derive the desired

result

Remark 21 We note that if the coefficients do not dependexplicitly on the marginal law of the underlying diffusionthe result reduces to the classical case that is the SMP forBSPDEs without mean-field term

Remark 22 When we remove the initial state constraint (75)we obtain the general maximum principle for the mean-fieldBSPDEs system (ie without the constraint) with 119897

1= 1

54 Application A Backward Linear Quadratic Control Prob-lem Now we apply our maximum principle to solve an LQproblem For notational simplicity we restrict ourselves tothe free case (ie without the initial state constraint (75)) thegeneral case being handled in a similar way


Consider the following problem

119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)

where 119860 is a partial differential operator satisfying condition(L1) and 119861 119861 119862119863119863 and119873 are bounded and deterministicconstantsWe also assume that119873 gt 0 and V isin 1198712F (0 119879 119880) It iseasy to verify that BSPDE (132) admits a uniquemild solution(119884(119905) 119885(119905))

119875(119905) the adjoint process of state equation (132) is thesolution of

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)

Let 119906(sdot) be an optimal control and let (119884(sdot) 119885(sdot)) bethe corresponding state process By maximum principle ofTheorem 20 (note that 119897

1= 1 in this problem)

1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)

for all V isin 119880ad since the state equation has the form (132)Thisin turn implies

119906 (119905) = minus119862

119873119875 (119905) (135)

It is clear that (131) is a positive quadratic functional of controlbecause of 119873 gt 0 Hence an optimal control exists Thecandidate optimal control (135) is indeed an optimal controlof this LQ problem for it is the only control which satisfies themaximum principle

Next we want to obtain a more explicit representationof the optimal control (135) from the state equation (132)Substituting (135) into state equation (132) yields

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)

Combining the above equation with (133) we obtain thefollowing related feedback control system

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)

Looking at the terminal condition of 119875(119905) in (133) andconsidering the mean-field type of (132) it is reasonable toconjecture that 119875(119905) has the following form

119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)

where120593(119905)120601(119905) are deterministic differential functionswhichwill be specified below Moreover 120593(0) = 1 120601(0) = 0

Inserting this form into adjoint equation (133) and notic-ing that 119884(119905) satisfies (136) we can compare the coefficientsof 119889119905 and 119889119882(119905) to obtain the following equation

2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)

Then subtracting 119885(119905) we have

2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)

Comparing the coefficients of 119884(119905) and E[119884(119905)] respec-tively we get

119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)

We solve (141) to get

120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)

Then (142) exists a unique solution from the classical Riccatiequation theory

We now conclude the above discussions in the followingresult

Theorem 23 For onersquos linear quadratic stochastic partialdifferential control problem (131)-(132) the unique optimalcontrol 119906(sdot) isin 119880ad is given by

119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)

with 120593(119905) satisfying (143) and 120601(119905) solving (142)

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgment

This research is partially supported by the Natural ScienceFoundation of China under Grant no 11301310

References

[1] Y Hu and S Peng ldquoAdapted solution of a backward semilinearstochastic evolution equationrdquo Stochastic Analysis and Applica-tions vol 9 no 4 pp 445ndash459 1991

[2] N IMahmudov andMAMcKibben ldquoOnbackward stochasticevolution equations in Hilbert spaces and optimal controlrdquoNonlinear Analysis vol 67 no 4 pp 1260ndash1274 2007

[3] M Fuhrman and G Tessitore ldquoNonlinear Kolmogorov equa-tions in infinite dimensional spaces the backward stochasticdifferential equations approach and applications to optimalcontrolrdquoThe Annals of Probability vol 30 no 3 pp 1397ndash14652002

[4] M Fuhrman and G Tessitore ldquoInfinite horizon backwardstochastic differential equations and elliptic equations inHilbert spacesrdquo Annals of Probability vol 32 no 1B pp 607ndash660 2004

[5] X Mao ldquoAdapted solutions of backward stochastic differentialequations with non-Lipschitz coefficientsrdquo Stochastic Processesand their Applications vol 58 no 2 pp 281ndash292 1995

[6] J Lasry and P Lions ldquoMean field gamesrdquo Japanese Journal ofMathematics vol 2 no 1 pp 229ndash260 2007

[7] R Buckdahn J Li and S Peng ldquoMean-field backward stochas-tic differential equations and related partial differential equa-tionsrdquo Stochastic Processes and their Applications vol 119 no 10pp 3133ndash3154 2009

[8] D Andersson and B Djehiche ldquoAmaximum principle for SDEsofmean-field typerdquoAppliedMathematics andOptimization vol63 no 3 pp 341ndash356 2011

[9] R Buckdahn B Djehiche J Li and S Peng ldquoMean-fieldbackward stochastic differential equations a limit approachrdquoThe Annals of Probability vol 37 no 4 pp 1524ndash1565 2009

[10] R Xu ldquoMean-field backward doubly stochastic differentialequations and related SPDEsrdquo Boundary Value Problems vol2012 article 114 2012

[11] R Buckdahn B Djehiche and J Li ldquoA general stochasticmaximum principle for SDEs of mean-field typerdquo AppliedMathematics and Optimization vol 64 no 2 pp 197ndash216 2011

[12] J Li ldquoStochasticmaximumprinciple in themean-field controlsrdquoAutomatica vol 48 no 2 pp 366ndash373 2012

[13] G Wang C Zhang and W Zhang ldquoStochastic maximumprinciple for mean-field type optimal control under partialinformationrdquo IEEE Transactions on Automatic Control vol 59no 2 pp 522ndash528 2014

[14] M Hafayed ldquoA mean-field necessary and sufficient conditionsfor optimal singular stochastic controlrdquo Communications inMathematics and Statistics vol 1 no 4 pp 417ndash435 2013

[15] M Hafayed ldquoA mean-field maximum principle for optimalcontrol of forward-backward stochastic differential equationswith Poisson jumpprocessesrdquo International Journal ofDynamicsand Control vol 1 no 4 pp 300ndash315 2013

[16] Y Hu and S Peng ldquoMaximum principle for semilinearstochastic evolution control systemsrdquo Stochastics and StochasticsReports vol 33 no 3-4 pp 159ndash180 1990

[17] M Fuhrman Y Hu and G Tessitore ldquoStochastic maximumprinciple for optimal control of SPDEsrdquo Comptes Rendus Math-ematique vol 350 no 13-14 pp 683ndash688 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In this paper we investigate a new type of backwardstochastic evolution equations inHilbert spaces whichwe callmean-field BSEEs Mean-field implies that the coefficient ofthe BSEE depends on the law of the BSEE Specifically theBSEE we consider is defined as

119889119884 (119904) = minus 119860119884 (119904) 119889119904

minus E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

+ 119885 (119904) 119889119882 (119904)

119884 (119879) = 120585 119904 isin [0 119879]

(1)

in a Hilbert space 119867 where 119891 denotes a given measurablemapping 119879 is a fixed positive real number 119882(119904) is acylindrical Wiener process and 119860 represents the generatorof a strongly continuous semigroup 119890

119905119860 in 119867 with 119905 ge 0Precise interpretation of E1015840[119891(119904 1198841015840

(119904) 1198851015840(119904) 119884(119904) 119885(119904))] is

given in the following sections Based on the contractionmapping we firstly prove that (1) admits a unique mildsolution if the function 119891 is Lipschitz continuous Secondlyunder non-Lipschitz assumptions we obtain the existenceand uniqueness of the mild solution for mean-field BSEEby constructing a special Cauchy sequence The Lipschitzcondition is a special case of this non-Lipschitz condition (seeMao [5]) In addition we investigate the well-posedness ofmean-field stochastic evolution equations

We also study optimal control problems of stochasticsystems governed by mean-field BSPDEs in Hilbert spacesOur objective is to formulate a stochastic maximumprinciple(SMP) for the optimal control problem with an initial stateconstraint There is a vast literature on the theory of SMPAmong these papers Andersson and Djehiche [8] studiedthe optimal control problem for mean-field stochastic systemwhen the control domain is convex They obtained themaximum principle by a convex variational method By aspike variational technique Buckdahn et al [11] obtained ageneral maximum principle for a special mean-field stochas-tic differential equation (SDE) where the action space is notconvex Later Li [12] investigated the maximum principlefor more general SDEs of mean-field type with a convexcontrol domain Wang et al [13] were concerned with apartially observed optimal control problem of mean-fieldtype By using Girsanovrsquos theorem and convex variationthey derived the correspondingmaximum principle and gavean illustrative example to demonstrate the application ofthe obtained SMP Hafayed studied the mean-field SMP forsingular stochastic control in [14] and mean-field SMP forFBSDEs with Poisson jump processes in [15]

For the case of stochastic control systems in infinitedimensions on the assumption that the control domain isnot necessarily convex while the diffusion coefficient doesnot contain the control variable Hu and Peng [16] usedspike variation approach and Ekelandrsquos variational principleto establish the maximum principle for semilinear stochas-tic evolution control systems with a final state constraintMahmudov and Mckibben [2] obtained an SMP for stochas-tic control systems governed by BSEEs in Hilbert spacesRecently Fuhrman et al [17] deduced themaximumprinciple

for optimal control of stochastic PDEs when the controldomain is not necessarily convex

We establish necessary optimality conditions for thecontrol problem in the form of a maximum principle on theassumption that the control domain is not necessarily convexDue to the initial state constraint we first need to applyEkelandrsquos variational principle to convert the given controlproblem into a free initial state optimal control problemThenspike variation approach is used to deduce the SMP in themean-field framework In our control system not only thestate processes which are the unique mild solution of thegiven BSPDE but also the cost functional are of mean-fieldtype In other words they depend on the law of the BSPDEas well as the state and the control For this new controlledsystem the adjoint equation will turn out to be a mean-fieldstochastic evolution equation

Theplan of this paper is organized as follows In Section 2we introduce some notations which are needed in whatfollows In Section 3 the well-posedness of mean-field BSEE(1) is studied we first prove the existence and uniqueness of amild solution under the Lipschitz condition and investigatethe regular dependence of the solution (119884 119885) on (120585 119891)And then under the assumption that the coefficient is non-Lipschitz continuous a new result on the existence anduniqueness of the mild solution to (1) in Hilbert space isestablished which generalizes the result for the Lipschitzcase Section 4 is devoted to the regularity of mean-fieldstochastic evolution equations In Section 5 we derive thestochastic maximum principle for the BSPDE systems ofmean-field type with an initial state constraint and at thelast part of Section 5 an LQ example is given to show theapplication of our maximum principle An explicit optimalcontrol is obtained in this example

2 Preliminaries

The norm of an element 119909 in a Banach space 119865 is denotedby |119909|

119865or simply |119909| if no confusion is possible Γ 119867 and

119870 are three real and separable Hilbert spaces Scalar productis denoted by ⟨sdot sdot⟩ with a subscript to specify the space ifnecessaryL(Γ 119870) is the space of Hilbert-Schmidt operatorsfrom Γ to 119870 endowed with the Hilbert-Schmidt norm

Let (ΩFP) be a complete probability space A cylin-drical Wiener process 119882(119905) 119905 ge 0 in a Hilbert space Γ is afamily of linear mappings Γ rarr 119871

2(ΩFP) such that

(i) for every 119906 isin Γ 119882(119905)119906 119905 ge 0 is a real (continuous)Wiener process

(ii) for every 119906 V isin Γ and 119905 119904 ge 0 E(119882(119905)119906 sdot 119882(119904)V) =(119905 and 119904)⟨119906 V⟩

Γ

By F119905 119905 isin [0 119879] we denote the natural filtration of 119882

augmented with the familyN of P-null sets ofF119879

F119905= 120590 (119882 (119904) 119904 isin [0 119905]) orN (2)

The filtration (F119905)119905ge0

satisfies the usual conditions Allthe concepts of measurability for stochastic processes (egadapted etc) refer to this filtration



(I) H2




(i) Eint1198790|120595

119905|2119889119905 lt +infin

(ii) 120595119905isF


Evidently H2


10038171003817100381710038171205951003817100381710038171003817 = Eint

119879

0

100381610038161003816100381612059511990410038161003816100381610038162

119889119904

12

(3)

(II) S2



(i) E(sup0le119905le119879

|120595119905|2) lt +infin

(ii) 120595119905isF





1003817100381710038171003817(119910 119911)10038171003817100381710038172

120573119905= Eint

119879

119905

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904 (4)

on the Banach space

K120573[119905 119879] = S

2

F ([119905 119879] 119867) timesH2

F ([119905 119879] L (Γ119867)) (5)




with the norm

1003817100381710038171003817(119910 119911)10038171003817100381710038172

= Eint119879

0

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904 (6)





119879P 119867) times

H2


119884 (119905) = 119878 (119879 minus 119905) 120585 + int

119879

119905

119878 (119904 minus 119905) 119891 (119904) 119889119904

+ int

119879

119905

119878 (119904 minus 119905) 119885 (119904) 119889119882 (119904) 119875-119886119904(7)


E sup119905le119904le119879

1198902120573119904

|119884 (119904)|2+ Eint

119879

119905

1198902120573119904

|119885 (119904)|2119889119904

le 241198722

119878(119890

2120573119879E100381610038161003816100381612058510038161003816100381610038162

+1

2120573int

119879

119905

1198902120573119903

E1003816100381610038161003816119891 (119903)

10038161003816100381610038162

119889119903)

(8)






119905= F otimes F




1015840 120596) =

120585(1205961015840) (120596


1(ΩFP) the


E1015840[120578 (sdot 120596)] = int

Ω

120578 (1205961015840 120596)P (119889120596

1015840) (9)


E [120578] (= intΩ

120578119889P = intΩ

E1015840[120578 (sdot 120596)]P (119889120596)) = E [E

1015840[120578]]

(10)



119889119884 (119904) = minus 119860119884 (119904) 119889119904

minus E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

+ 119885 (119904) 119889119882 (119904)

119884 (119879) = 120585 119904 isin [0 119879]

(11)


119860≜ sup

119905isin[0119879]|119890119905119860|



S2

F ([0 119879]119867) timesH2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

119890119860(119904minus119905)

E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(12)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] (120596)

= E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (120596 119904) 119885 (120596 119904))]

= intΩ

119891 (1205961015840 120596 119904 119884 (120596

1015840 119904) 119885 (120596

1015840 119904) 119884 (120596 119904) 119885 (120596 119904))

times P (1198891205961015840)

(13)



10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 119871 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

+100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(14)

for all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 119911

1015840

119894 119911

119894isin L(Γ119867) (119894 =

1 2)(A2) 119891(sdot 0 0 0 0) isin H2



119879P 119867)


F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867))



F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867))BSEE119884 (119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904) 0 le 119905 le 119879

(15)


119891(119910119911)

(119904 120583 ]) = E1015840[119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 120583 ])] (16)


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

119890119860(119904minus119905)

119891(119910119911)

(119884 (119904) 119885 (119904)) 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(17)


1) (120583

2 ]


10038161003816100381610038161003816119891(119910119911)

(1205831 ]

1) minus 119891

(119910119911)(120583

2 ]

2)10038161003816100381610038161003816

2

le 119871 (10038161003816100381610038161205831 minus 1205832

10038161003816100381610038162

+1003816100381610038161003816]1 minus ]

2

10038161003816100381610038162

)

(18)



Φ[(1199101015840(sdot) 119911


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904) 0 le 119905 le 119879

(19)


119894) = Φ[(119910

119894 119911

119894)] 119894 =

1 2 (119910 119911) = (1199101minus119910

2 119911

1minus119911

2) and (119884 119885) = (119884

1minus119884

2 119885

1minus119885

2)


E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E


times [int

119879

0

1198902120573119904

100381610038161003816100381610038161003816E1015840[119891 (119904 (119910

1

(119904))1015840

(1199111

(119904))1015840

1198841

(119904) 1198851

(119904))

minus 119891 (119904 (1199102

(119904))1015840

(1199112

(119904))1015840

1198842

(119904) 1198852

(119904))]100381610038161003816100381610038161003816

2

119889119904]

le12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(E [1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2]

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(20)

If we set 120573 = 361198722

1198601198712max119879 1 then

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le 119879 sdot E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

1198601198712max 119879 1120573

E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=1

3E [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(21)

That is

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le1

2Eint

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904

(22)


120573[0 119879] with the norm

(119884 119885)2

120573= Eint

119879

0

1198902120573119904

(|119884 (119904)|2+ |119885 (119904)|

2) 119889119904 (23)


120573[0 119879] such that Φ(119884 119885) =


F ([0 119879]119867) times

H2





E1015840[119891

120572(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

997888rarr E1015840[119891

1205720(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

(24)


F ([0 119879]119867) times

H2



1198712(ΩF



119879P 119867) to S2

F ([0 119879]119867) times

H2

F ([0 119879]L(Γ119867))


(A3) For all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 1199111015840

119894 119911

119894isin L(Γ119867) (119894 =


10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 120579 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

) + 120579 (10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

)

+ 119897 (100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(25)





120579 (119906) le 119886 + 119887119906 (26)


F ([0 119879]119867) whenever


1199101015840(sdot) 119910(sdot) isin S2


F ([0 119879]L(Γ119867))


119899 119885


S2

F ([0 119879]119867) timesH2


119884119899(119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)

times119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885119899(119904) 119889119882 (119904)

(27)


119899(119905) 119885

119899(119905))


119899(119905) 119885

119899(119905)) 119905 isin

[0 119879]


1and 119862

2such that

(i)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905)

(ii)Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904 le 21198621exp (119879 minus 119905)

(iii)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le 1198622int

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(28)



yields

10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

le 2120579 (100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 2120579 (1003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

)

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

le 4119886 + 2119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 21198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

(29)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) + Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))]

10038161003816100381610038161003816

2

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904 10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le 1198621+24119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904[119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 119897100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

]] 119889119904

= 1198621+48119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

(30)

where

1198621= 24119872

2

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[2119886 +1003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

] 119889119904 + 1

(31)

If we set 120573 = 961198722

119860max119887 119897 we can obtain

sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) +1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

(32)


sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905

2)

le 21198621exp (119879 minus 119905)

(33)




int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 21198621+ int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

le 21198621+ 2119862

1int

119879

119905

exp (119879 minus 119904) 119889119904

= 21198621exp (119879 minus 119905)

(34)


E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

]

le E1015840[120579 (

100381610038161003816100381610038161198841015840

119899(119904) minus 119884

1015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 119897100381610038161003816100381610038161198851015840

119899+1(119904) minus 119885

1015840

119899(119904)10038161003816100381610038161003816

2

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

= E [120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

(35)

By Lemma 1 we have

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

+ Eint119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))]

10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904

E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)

+ 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

] 119889119904

(36)


(1 minus24119872

2

119860119897

120573)Eint

119879

119905

11989021205731199041003817100381710038171003817119885119899+1

(119904) minus 119885119899(119904)10038171003817100381710038172

119889119904 ge 0 (37)

Then

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(38)

where we set 1198622= (24119872

2

119860120573)119890

2120573119879



119898= 119879 by setting 120591

119896= 119896120575 119896 = 1 2 3 119898 with

120575 = 119879119898


120591119896] define

1198623= 119862

2120579 (2119862

1exp (119879))

1205931198961(119905) = 119862

3(120591119896minus 119905)

120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 119899 ge 1

(39)


0 le 120593119896119899(119905) le 120593


1198961(119905) (40)


120591119896]


1205931198962(119905) = 119862

2int

120591119896

119905

120579 (1205931198961(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (1198623(120591119896minus 119904)) 119889119904

le 1198623(120591119896minus 119905) = 120593

1198961(119905)

(41)


1198622120579 (119862

3(120591119896minus 119905)) le 119862

3= 119862

2120579 (2119862

1exp (119879)) (42)


or

1198623(120591119896minus 119905) = 119862

2120579 (2119862

1exp (119879)) (120591

119896minus 119905) le 2119862

1exp (119879)

(43)


if

1198622(119886 + 119887) (120591

119896minus 119905) le 1 (44)


120591119896] 119896 = 1 2 119898 if 120591

119896minus

120591119896minus1





120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 le 119862

2int

120591119896

119905

120579 (120593119896119899minus1

(119904)) 119889119904

= 120593119896119899(119905) forall119905 isin [120591

119896minus1 120591

119896]

(45)






E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) + Eint119879

119905

119890212057311990410038161003816100381610038161003816

119885 (119904) minus 119885 (119904)10038161003816100381610038161003816

2

119889119904

le24119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

) + 11989710038161003816100381610038161003816119885 (119904) minus 119885 (119904)

10038161003816100381610038161003816

2

] 119889119904

(46)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

)] 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

119890212057311990310038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)119889119904

(47)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) = 0 (48)



119899) defined by (27)

satisfies

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

997888rarr 0 forall0 le 119905 le 119879 (49)


119896minus1 120591

119896] we set 120593

119896119899(119905) =

E sup119904isin[119905120591119896]

1198902120573119904|119884119899+1

(119904) minus 119884119899(119904)|

2 By Lemmas 6 and 7

1205931198961(119905) = E sup

119904isin[119905120591119896]

119890212057311990410038161003816100381610038161198842 (119904) minus 1198841 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119904le119903le120591119896

119890212057311990310038161003816100381610038161198841 (119903) minus 1198840 (119903)

10038161003816100381610038162

)119889119904

le 1198622int

120591119896

119905

120579 (21198621exp (120591

119896minus 119905)) 119889119904

le 1198622120579 (2119862

1exp (119879)) (120591

119896minus 119905) = 119862

3(120591119896minus 119905) = 120593

1198961(119905)

(50)

Suppose that 120593119896119899(119905) le 120593

119896119899(119905) holds for some 119899 ge 1


120591119896]

we obtain

120593119896119899+1

(119905) = E sup119904isin[119905120591119896]

11989021205731199041003816100381610038161003816119884119899+2 (119904) minus 119884119899+1 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119903isin[119904120591119896]

11989021205731199031003816100381610038161003816119884119899+1 (119903) minus 119884119899 (119903)

10038161003816100381610038162

)119889119904

= 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904

le 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 120593

119896119899+1(119905)

(51)


120593119896119899(119905) le 120593

119896119899(119905) (52)


119896minus1 120591

119896] Note that


119896minus1 120591



the function 120593119896(119905) by 120593

119896119899(119905) darr 120593



119896minus1 120591

119896] By the


119896(119905) we get

120593119896(119905) = lim

119899rarrinfin

1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (120593119896(119904)) 119889119904

(53)


le 119905 le 120591119896 Since int

0+(119889119906120579(119906)) = infin the Bihari

inequality implies

120593119896(119905) = 0 for each 119905 isin [120591

119896minus1 120591

119896] (54)


lim119899rarrinfin

120593119896119899(119905) = 0 (55)


Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)


119899) is


F ([0 119879]119867) times

H2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)





119879-measurable


(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)


119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))


2

(O))


2

(O)) 997888rarr R

(61)


(H1) 119886119894119895 119887

119894 R119899


119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|




1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)


2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))


119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)



119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(I) H2




(i) Eint1198790|120595

119905|2119889119905 lt +infin

(ii) 120595119905isF


Evidently H2


10038171003817100381710038171205951003817100381710038171003817 = Eint

119879

0

100381610038161003816100381612059511990410038161003816100381610038162

119889119904

12

(3)

(II) S2



(i) E(sup0le119905le119879

|120595119905|2) lt +infin

(ii) 120595119905isF





1003817100381710038171003817(119910 119911)10038171003817100381710038172

120573119905= Eint

119879

119905

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904 (4)

on the Banach space

K120573[119905 119879] = S

2

F ([119905 119879] 119867) timesH2

F ([119905 119879] L (Γ119867)) (5)




with the norm

1003817100381710038171003817(119910 119911)10038171003817100381710038172

= Eint119879

0

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904 (6)





119879P 119867) times

H2


119884 (119905) = 119878 (119879 minus 119905) 120585 + int

119879

119905

119878 (119904 minus 119905) 119891 (119904) 119889119904

+ int

119879

119905

119878 (119904 minus 119905) 119885 (119904) 119889119882 (119904) 119875-119886119904(7)


E sup119905le119904le119879

1198902120573119904

|119884 (119904)|2+ Eint

119879

119905

1198902120573119904

|119885 (119904)|2119889119904

le 241198722

119878(119890

2120573119879E100381610038161003816100381612058510038161003816100381610038162

+1

2120573int

119879

119905

1198902120573119903

E1003816100381610038161003816119891 (119903)

10038161003816100381610038162

119889119903)

(8)






119905= F otimes F




1015840 120596) =

120585(1205961015840) (120596


1(ΩFP) the


E1015840[120578 (sdot 120596)] = int

Ω

120578 (1205961015840 120596)P (119889120596

1015840) (9)


E [120578] (= intΩ

120578119889P = intΩ

E1015840[120578 (sdot 120596)]P (119889120596)) = E [E

1015840[120578]]

(10)



119889119884 (119904) = minus 119860119884 (119904) 119889119904

minus E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

+ 119885 (119904) 119889119882 (119904)

119884 (119879) = 120585 119904 isin [0 119879]

(11)


119860≜ sup

119905isin[0119879]|119890119905119860|



S2

F ([0 119879]119867) timesH2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

119890119860(119904minus119905)

E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(12)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] (120596)

= E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (120596 119904) 119885 (120596 119904))]

= intΩ

119891 (1205961015840 120596 119904 119884 (120596

1015840 119904) 119885 (120596

1015840 119904) 119884 (120596 119904) 119885 (120596 119904))

times P (1198891205961015840)

(13)



10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 119871 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

+100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(14)

for all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 119911

1015840

119894 119911

119894isin L(Γ119867) (119894 =

1 2)(A2) 119891(sdot 0 0 0 0) isin H2



119879P 119867)


F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867))



F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867))BSEE119884 (119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904) 0 le 119905 le 119879

(15)


119891(119910119911)

(119904 120583 ]) = E1015840[119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 120583 ])] (16)


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

119890119860(119904minus119905)

119891(119910119911)

(119884 (119904) 119885 (119904)) 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(17)


1) (120583

2 ]


10038161003816100381610038161003816119891(119910119911)

(1205831 ]

1) minus 119891

(119910119911)(120583

2 ]

2)10038161003816100381610038161003816

2

le 119871 (10038161003816100381610038161205831 minus 1205832

10038161003816100381610038162

+1003816100381610038161003816]1 minus ]

2

10038161003816100381610038162

)

(18)



Φ[(1199101015840(sdot) 119911


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904) 0 le 119905 le 119879

(19)


119894) = Φ[(119910

119894 119911

119894)] 119894 =

1 2 (119910 119911) = (1199101minus119910

2 119911

1minus119911

2) and (119884 119885) = (119884

1minus119884

2 119885

1minus119885

2)


E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E


times [int

119879

0

1198902120573119904

100381610038161003816100381610038161003816E1015840[119891 (119904 (119910

1

(119904))1015840

(1199111

(119904))1015840

1198841

(119904) 1198851

(119904))

minus 119891 (119904 (1199102

(119904))1015840

(1199112

(119904))1015840

1198842

(119904) 1198852

(119904))]100381610038161003816100381610038161003816

2

119889119904]

le12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(E [1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2]

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(20)

If we set 120573 = 361198722

1198601198712max119879 1 then

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le 119879 sdot E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

1198601198712max 119879 1120573

E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=1

3E [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(21)

That is

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le1

2Eint

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904

(22)


120573[0 119879] with the norm

(119884 119885)2

120573= Eint

119879

0

1198902120573119904

(|119884 (119904)|2+ |119885 (119904)|

2) 119889119904 (23)


120573[0 119879] such that Φ(119884 119885) =


F ([0 119879]119867) times

H2





E1015840[119891

120572(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

997888rarr E1015840[119891

1205720(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

(24)


F ([0 119879]119867) times

H2



1198712(ΩF



119879P 119867) to S2

F ([0 119879]119867) times

H2

F ([0 119879]L(Γ119867))


(A3) For all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 1199111015840

119894 119911

119894isin L(Γ119867) (119894 =


10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 120579 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

) + 120579 (10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

)

+ 119897 (100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(25)





120579 (119906) le 119886 + 119887119906 (26)


F ([0 119879]119867) whenever


1199101015840(sdot) 119910(sdot) isin S2


F ([0 119879]L(Γ119867))


119899 119885


S2

F ([0 119879]119867) timesH2


119884119899(119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)

times119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885119899(119904) 119889119882 (119904)

(27)


119899(119905) 119885

119899(119905))


119899(119905) 119885

119899(119905)) 119905 isin

[0 119879]


1and 119862

2such that

(i)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905)

(ii)Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904 le 21198621exp (119879 minus 119905)

(iii)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le 1198622int

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(28)



yields

10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

le 2120579 (100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 2120579 (1003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

)

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

le 4119886 + 2119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 21198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

(29)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) + Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))]

10038161003816100381610038161003816

2

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904 10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le 1198621+24119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904[119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 119897100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

]] 119889119904

= 1198621+48119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

(30)

where

1198621= 24119872

2

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[2119886 +1003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

] 119889119904 + 1

(31)

If we set 120573 = 961198722

119860max119887 119897 we can obtain

sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) +1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

(32)


sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905

2)

le 21198621exp (119879 minus 119905)

(33)




int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 21198621+ int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

le 21198621+ 2119862

1int

119879

119905

exp (119879 minus 119904) 119889119904

= 21198621exp (119879 minus 119905)

(34)


E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

]

le E1015840[120579 (

100381610038161003816100381610038161198841015840

119899(119904) minus 119884

1015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 119897100381610038161003816100381610038161198851015840

119899+1(119904) minus 119885

1015840

119899(119904)10038161003816100381610038161003816

2

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

= E [120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

(35)

By Lemma 1 we have

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

+ Eint119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))]

10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904

E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)

+ 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

] 119889119904

(36)


(1 minus24119872

2

119860119897

120573)Eint

119879

119905

11989021205731199041003817100381710038171003817119885119899+1

(119904) minus 119885119899(119904)10038171003817100381710038172

119889119904 ge 0 (37)

Then

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(38)

where we set 1198622= (24119872

2

119860120573)119890

2120573119879



119898= 119879 by setting 120591

119896= 119896120575 119896 = 1 2 3 119898 with

120575 = 119879119898


120591119896] define

1198623= 119862

2120579 (2119862

1exp (119879))

1205931198961(119905) = 119862

3(120591119896minus 119905)

120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 119899 ge 1

(39)


0 le 120593119896119899(119905) le 120593


1198961(119905) (40)


120591119896]


1205931198962(119905) = 119862

2int

120591119896

119905

120579 (1205931198961(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (1198623(120591119896minus 119904)) 119889119904

le 1198623(120591119896minus 119905) = 120593

1198961(119905)

(41)


1198622120579 (119862

3(120591119896minus 119905)) le 119862

3= 119862

2120579 (2119862

1exp (119879)) (42)


or

1198623(120591119896minus 119905) = 119862

2120579 (2119862

1exp (119879)) (120591

119896minus 119905) le 2119862

1exp (119879)

(43)


if

1198622(119886 + 119887) (120591

119896minus 119905) le 1 (44)


120591119896] 119896 = 1 2 119898 if 120591

119896minus

120591119896minus1





120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 le 119862

2int

120591119896

119905

120579 (120593119896119899minus1

(119904)) 119889119904

= 120593119896119899(119905) forall119905 isin [120591

119896minus1 120591

119896]

(45)






E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) + Eint119879

119905

119890212057311990410038161003816100381610038161003816

119885 (119904) minus 119885 (119904)10038161003816100381610038161003816

2

119889119904

le24119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

) + 11989710038161003816100381610038161003816119885 (119904) minus 119885 (119904)

10038161003816100381610038161003816

2

] 119889119904

(46)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

)] 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

119890212057311990310038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)119889119904

(47)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) = 0 (48)



119899) defined by (27)

satisfies

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

997888rarr 0 forall0 le 119905 le 119879 (49)


119896minus1 120591

119896] we set 120593

119896119899(119905) =

E sup119904isin[119905120591119896]

1198902120573119904|119884119899+1

(119904) minus 119884119899(119904)|

2 By Lemmas 6 and 7

1205931198961(119905) = E sup

119904isin[119905120591119896]

119890212057311990410038161003816100381610038161198842 (119904) minus 1198841 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119904le119903le120591119896

119890212057311990310038161003816100381610038161198841 (119903) minus 1198840 (119903)

10038161003816100381610038162

)119889119904

le 1198622int

120591119896

119905

120579 (21198621exp (120591

119896minus 119905)) 119889119904

le 1198622120579 (2119862

1exp (119879)) (120591

119896minus 119905) = 119862

3(120591119896minus 119905) = 120593

1198961(119905)

(50)

Suppose that 120593119896119899(119905) le 120593

119896119899(119905) holds for some 119899 ge 1


120591119896]

we obtain

120593119896119899+1

(119905) = E sup119904isin[119905120591119896]

11989021205731199041003816100381610038161003816119884119899+2 (119904) minus 119884119899+1 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119903isin[119904120591119896]

11989021205731199031003816100381610038161003816119884119899+1 (119903) minus 119884119899 (119903)

10038161003816100381610038162

)119889119904

= 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904

le 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 120593

119896119899+1(119905)

(51)


120593119896119899(119905) le 120593

119896119899(119905) (52)


119896minus1 120591

119896] Note that


119896minus1 120591



the function 120593119896(119905) by 120593

119896119899(119905) darr 120593



119896minus1 120591

119896] By the


119896(119905) we get

120593119896(119905) = lim

119899rarrinfin

1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (120593119896(119904)) 119889119904

(53)


le 119905 le 120591119896 Since int

0+(119889119906120579(119906)) = infin the Bihari

inequality implies

120593119896(119905) = 0 for each 119905 isin [120591

119896minus1 120591

119896] (54)


lim119899rarrinfin

120593119896119899(119905) = 0 (55)


Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)


119899) is


F ([0 119879]119867) times

H2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)





119879-measurable


(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)


119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))


2

(O))


2

(O)) 997888rarr R

(61)


(H1) 119886119894119895 119887

119894 R119899


119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|




1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)


2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))


119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)



119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











S2

F ([0 119879]119867) timesH2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

119890119860(119904minus119905)

E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(12)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] (120596)

= E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (120596 119904) 119885 (120596 119904))]

= intΩ

119891 (1205961015840 120596 119904 119884 (120596

1015840 119904) 119885 (120596

1015840 119904) 119884 (120596 119904) 119885 (120596 119904))

times P (1198891205961015840)

(13)



10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 119871 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

+100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(14)

for all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 119911

1015840

119894 119911

119894isin L(Γ119867) (119894 =

1 2)(A2) 119891(sdot 0 0 0 0) isin H2



119879P 119867)


F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867))



F ([0 119879]119867) timesH2

F ([0 119879]L(Γ119867))BSEE119884 (119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904) 0 le 119905 le 119879

(15)


119891(119910119911)

(119904 120583 ]) = E1015840[119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 120583 ])] (16)


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

119890119860(119904minus119905)

119891(119910119911)

(119884 (119904) 119885 (119904)) 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(17)


1) (120583

2 ]


10038161003816100381610038161003816119891(119910119911)

(1205831 ]

1) minus 119891

(119910119911)(120583

2 ]

2)10038161003816100381610038161003816

2

le 119871 (10038161003816100381610038161205831 minus 1205832

10038161003816100381610038162

+1003816100381610038161003816]1 minus ]

2

10038161003816100381610038162

)

(18)



Φ[(1199101015840(sdot) 119911


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119910

1015840

(119904) 1199111015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904) 0 le 119905 le 119879

(19)


119894) = Φ[(119910

119894 119911

119894)] 119894 =

1 2 (119910 119911) = (1199101minus119910

2 119911

1minus119911

2) and (119884 119885) = (119884

1minus119884

2 119885

1minus119885

2)


E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E


times [int

119879

0

1198902120573119904

100381610038161003816100381610038161003816E1015840[119891 (119904 (119910

1

(119904))1015840

(1199111

(119904))1015840

1198841

(119904) 1198851

(119904))

minus 119891 (119904 (1199102

(119904))1015840

(1199112

(119904))1015840

1198842

(119904) 1198852

(119904))]100381610038161003816100381610038161003816

2

119889119904]

le12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(E [1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2]

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(20)

If we set 120573 = 361198722

1198601198712max119879 1 then

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le 119879 sdot E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

1198601198712max 119879 1120573

E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=1

3E [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(21)

That is

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le1

2Eint

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904

(22)


120573[0 119879] with the norm

(119884 119885)2

120573= Eint

119879

0

1198902120573119904

(|119884 (119904)|2+ |119885 (119904)|

2) 119889119904 (23)


120573[0 119879] such that Φ(119884 119885) =


F ([0 119879]119867) times

H2





E1015840[119891

120572(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

997888rarr E1015840[119891

1205720(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

(24)


F ([0 119879]119867) times

H2



1198712(ΩF



119879P 119867) to S2

F ([0 119879]119867) times

H2

F ([0 119879]L(Γ119867))


(A3) For all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 1199111015840

119894 119911

119894isin L(Γ119867) (119894 =


10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 120579 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

) + 120579 (10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

)

+ 119897 (100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(25)





120579 (119906) le 119886 + 119887119906 (26)


F ([0 119879]119867) whenever


1199101015840(sdot) 119910(sdot) isin S2


F ([0 119879]L(Γ119867))


119899 119885


S2

F ([0 119879]119867) timesH2


119884119899(119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)

times119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885119899(119904) 119889119882 (119904)

(27)


119899(119905) 119885

119899(119905))


119899(119905) 119885

119899(119905)) 119905 isin

[0 119879]


1and 119862

2such that

(i)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905)

(ii)Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904 le 21198621exp (119879 minus 119905)

(iii)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le 1198622int

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(28)



yields

10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

le 2120579 (100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 2120579 (1003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

)

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

le 4119886 + 2119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 21198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

(29)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) + Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))]

10038161003816100381610038161003816

2

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904 10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le 1198621+24119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904[119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 119897100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

]] 119889119904

= 1198621+48119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

(30)

where

1198621= 24119872

2

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[2119886 +1003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

] 119889119904 + 1

(31)

If we set 120573 = 961198722

119860max119887 119897 we can obtain

sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) +1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

(32)


sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905

2)

le 21198621exp (119879 minus 119905)

(33)




int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 21198621+ int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

le 21198621+ 2119862

1int

119879

119905

exp (119879 minus 119904) 119889119904

= 21198621exp (119879 minus 119905)

(34)


E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

]

le E1015840[120579 (

100381610038161003816100381610038161198841015840

119899(119904) minus 119884

1015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 119897100381610038161003816100381610038161198851015840

119899+1(119904) minus 119885

1015840

119899(119904)10038161003816100381610038161003816

2

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

= E [120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

(35)

By Lemma 1 we have

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

+ Eint119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))]

10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904

E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)

+ 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

] 119889119904

(36)


(1 minus24119872

2

119860119897

120573)Eint

119879

119905

11989021205731199041003817100381710038171003817119885119899+1

(119904) minus 119885119899(119904)10038171003817100381710038172

119889119904 ge 0 (37)

Then

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(38)

where we set 1198622= (24119872

2

119860120573)119890

2120573119879



119898= 119879 by setting 120591

119896= 119896120575 119896 = 1 2 3 119898 with

120575 = 119879119898


120591119896] define

1198623= 119862

2120579 (2119862

1exp (119879))

1205931198961(119905) = 119862

3(120591119896minus 119905)

120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 119899 ge 1

(39)


0 le 120593119896119899(119905) le 120593


1198961(119905) (40)


120591119896]


1205931198962(119905) = 119862

2int

120591119896

119905

120579 (1205931198961(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (1198623(120591119896minus 119904)) 119889119904

le 1198623(120591119896minus 119905) = 120593

1198961(119905)

(41)


1198622120579 (119862

3(120591119896minus 119905)) le 119862

3= 119862

2120579 (2119862

1exp (119879)) (42)


or

1198623(120591119896minus 119905) = 119862

2120579 (2119862

1exp (119879)) (120591

119896minus 119905) le 2119862

1exp (119879)

(43)


if

1198622(119886 + 119887) (120591

119896minus 119905) le 1 (44)


120591119896] 119896 = 1 2 119898 if 120591

119896minus

120591119896minus1





120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 le 119862

2int

120591119896

119905

120579 (120593119896119899minus1

(119904)) 119889119904

= 120593119896119899(119905) forall119905 isin [120591

119896minus1 120591

119896]

(45)






E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) + Eint119879

119905

119890212057311990410038161003816100381610038161003816

119885 (119904) minus 119885 (119904)10038161003816100381610038161003816

2

119889119904

le24119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

) + 11989710038161003816100381610038161003816119885 (119904) minus 119885 (119904)

10038161003816100381610038161003816

2

] 119889119904

(46)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

)] 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

119890212057311990310038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)119889119904

(47)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) = 0 (48)



119899) defined by (27)

satisfies

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

997888rarr 0 forall0 le 119905 le 119879 (49)


119896minus1 120591

119896] we set 120593

119896119899(119905) =

E sup119904isin[119905120591119896]

1198902120573119904|119884119899+1

(119904) minus 119884119899(119904)|

2 By Lemmas 6 and 7

1205931198961(119905) = E sup

119904isin[119905120591119896]

119890212057311990410038161003816100381610038161198842 (119904) minus 1198841 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119904le119903le120591119896

119890212057311990310038161003816100381610038161198841 (119903) minus 1198840 (119903)

10038161003816100381610038162

)119889119904

le 1198622int

120591119896

119905

120579 (21198621exp (120591

119896minus 119905)) 119889119904

le 1198622120579 (2119862

1exp (119879)) (120591

119896minus 119905) = 119862

3(120591119896minus 119905) = 120593

1198961(119905)

(50)

Suppose that 120593119896119899(119905) le 120593

119896119899(119905) holds for some 119899 ge 1


120591119896]

we obtain

120593119896119899+1

(119905) = E sup119904isin[119905120591119896]

11989021205731199041003816100381610038161003816119884119899+2 (119904) minus 119884119899+1 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119903isin[119904120591119896]

11989021205731199031003816100381610038161003816119884119899+1 (119903) minus 119884119899 (119903)

10038161003816100381610038162

)119889119904

= 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904

le 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 120593

119896119899+1(119905)

(51)


120593119896119899(119905) le 120593

119896119899(119905) (52)


119896minus1 120591

119896] Note that


119896minus1 120591



the function 120593119896(119905) by 120593

119896119899(119905) darr 120593



119896minus1 120591

119896] By the


119896(119905) we get

120593119896(119905) = lim

119899rarrinfin

1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (120593119896(119904)) 119889119904

(53)


le 119905 le 120591119896 Since int

0+(119889119906120579(119906)) = infin the Bihari

inequality implies

120593119896(119905) = 0 for each 119905 isin [120591

119896minus1 120591

119896] (54)


lim119899rarrinfin

120593119896119899(119905) = 0 (55)


Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)


119899) is


F ([0 119879]119867) times

H2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)





119879-measurable


(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)


119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))


2

(O))


2

(O)) 997888rarr R

(61)


(H1) 119886119894119895 119887

119894 R119899


119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|




1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)


2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))


119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)



119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times [int

119879

0

1198902120573119904

100381610038161003816100381610038161003816E1015840[119891 (119904 (119910

1

(119904))1015840

(1199111

(119904))1015840

1198841

(119904) 1198851

(119904))

minus 119891 (119904 (1199102

(119904))1015840

(1199112

(119904))1015840

1198842

(119904) 1198852

(119904))]100381610038161003816100381610038161003816

2

119889119904]

le12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(E [1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2]

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=12119872

2

1198601198712

120573E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(20)

If we set 120573 = 361198722

1198601198712max119879 1 then

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le 119879 sdot E sup0le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904)10038161003816100381610038161003816

2

+ Eint119879

0

119890212057311990410038161003816100381610038161003816

119885 (119904)10038161003816100381610038161003816

2

119889119904

le12119872

2

1198601198712max 119879 1120573

E

times [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

=1

3E [int

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2

+10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904]

(21)

That is

Eint119879

0

1198902120573119904

(10038161003816100381610038161003816119884 (119904)

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885 (119904)

10038161003816100381610038161003816

2

) 119889119904

le1

2Eint

119879

0

1198902120573119904

(1003816100381610038161003816119910 (119904)

10038161003816100381610038162

+ |119911 (119904)|2) 119889119904

(22)


120573[0 119879] with the norm

(119884 119885)2

120573= Eint

119879

0

1198902120573119904

(|119884 (119904)|2+ |119885 (119904)|

2) 119889119904 (23)


120573[0 119879] such that Φ(119884 119885) =


F ([0 119879]119867) times

H2





E1015840[119891

120572(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

997888rarr E1015840[119891

1205720(119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))]

(24)


F ([0 119879]119867) times

H2



1198712(ΩF



119879P 119867) to S2

F ([0 119879]119867) times

H2

F ([0 119879]L(Γ119867))


(A3) For all 119905 isin [0 119879] 1199101015840119894 119910

119894isin 119867 1199111015840

119894 119911

119894isin L(Γ119867) (119894 =


10038161003816100381610038161003816119891 (119905 119910

1015840

1 119911

1015840

1 119910

1 119911

1) minus 119891 (119905 119910

1015840

2 119911

1015840

2 119910

2 119911

2)10038161003816100381610038161003816

2

le 120579 (100381610038161003816100381610038161199101015840

1minus 119910

1015840

2

10038161003816100381610038161003816

2

) + 120579 (10038161003816100381610038161199101 minus 1199102

10038161003816100381610038162

)

+ 119897 (100381610038161003816100381610038161199111015840

1minus 119911

1015840

2

10038161003816100381610038161003816

2

+10038161003816100381610038161199111 minus 1199112

10038161003816100381610038162

)

(25)





120579 (119906) le 119886 + 119887119906 (26)


F ([0 119879]119867) whenever


1199101015840(sdot) 119910(sdot) isin S2


F ([0 119879]L(Γ119867))


119899 119885


S2

F ([0 119879]119867) timesH2


119884119899(119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)

times119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885119899(119904) 119889119882 (119904)

(27)


119899(119905) 119885

119899(119905))


119899(119905) 119885

119899(119905)) 119905 isin

[0 119879]


1and 119862

2such that

(i)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905)

(ii)Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904 le 21198621exp (119879 minus 119905)

(iii)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le 1198622int

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(28)



yields

10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

le 2120579 (100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 2120579 (1003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

)

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

le 4119886 + 2119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 21198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

(29)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) + Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))]

10038161003816100381610038161003816

2

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904 10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le 1198621+24119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904[119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 119897100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

]] 119889119904

= 1198621+48119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

(30)

where

1198621= 24119872

2

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[2119886 +1003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

] 119889119904 + 1

(31)

If we set 120573 = 961198722

119860max119887 119897 we can obtain

sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) +1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

(32)


sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905

2)

le 21198621exp (119879 minus 119905)

(33)




int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 21198621+ int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

le 21198621+ 2119862

1int

119879

119905

exp (119879 minus 119904) 119889119904

= 21198621exp (119879 minus 119905)

(34)


E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

]

le E1015840[120579 (

100381610038161003816100381610038161198841015840

119899(119904) minus 119884

1015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 119897100381610038161003816100381610038161198851015840

119899+1(119904) minus 119885

1015840

119899(119904)10038161003816100381610038161003816

2

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

= E [120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

(35)

By Lemma 1 we have

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

+ Eint119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))]

10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904

E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)

+ 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

] 119889119904

(36)


(1 minus24119872

2

119860119897

120573)Eint

119879

119905

11989021205731199041003817100381710038171003817119885119899+1

(119904) minus 119885119899(119904)10038171003817100381710038172

119889119904 ge 0 (37)

Then

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(38)

where we set 1198622= (24119872

2

119860120573)119890

2120573119879



119898= 119879 by setting 120591

119896= 119896120575 119896 = 1 2 3 119898 with

120575 = 119879119898


120591119896] define

1198623= 119862

2120579 (2119862

1exp (119879))

1205931198961(119905) = 119862

3(120591119896minus 119905)

120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 119899 ge 1

(39)


0 le 120593119896119899(119905) le 120593


1198961(119905) (40)


120591119896]


1205931198962(119905) = 119862

2int

120591119896

119905

120579 (1205931198961(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (1198623(120591119896minus 119904)) 119889119904

le 1198623(120591119896minus 119905) = 120593

1198961(119905)

(41)


1198622120579 (119862

3(120591119896minus 119905)) le 119862

3= 119862

2120579 (2119862

1exp (119879)) (42)


or

1198623(120591119896minus 119905) = 119862

2120579 (2119862

1exp (119879)) (120591

119896minus 119905) le 2119862

1exp (119879)

(43)


if

1198622(119886 + 119887) (120591

119896minus 119905) le 1 (44)


120591119896] 119896 = 1 2 119898 if 120591

119896minus

120591119896minus1





120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 le 119862

2int

120591119896

119905

120579 (120593119896119899minus1

(119904)) 119889119904

= 120593119896119899(119905) forall119905 isin [120591

119896minus1 120591

119896]

(45)






E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) + Eint119879

119905

119890212057311990410038161003816100381610038161003816

119885 (119904) minus 119885 (119904)10038161003816100381610038161003816

2

119889119904

le24119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

) + 11989710038161003816100381610038161003816119885 (119904) minus 119885 (119904)

10038161003816100381610038161003816

2

] 119889119904

(46)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

)] 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

119890212057311990310038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)119889119904

(47)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) = 0 (48)



119899) defined by (27)

satisfies

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

997888rarr 0 forall0 le 119905 le 119879 (49)


119896minus1 120591

119896] we set 120593

119896119899(119905) =

E sup119904isin[119905120591119896]

1198902120573119904|119884119899+1

(119904) minus 119884119899(119904)|

2 By Lemmas 6 and 7

1205931198961(119905) = E sup

119904isin[119905120591119896]

119890212057311990410038161003816100381610038161198842 (119904) minus 1198841 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119904le119903le120591119896

119890212057311990310038161003816100381610038161198841 (119903) minus 1198840 (119903)

10038161003816100381610038162

)119889119904

le 1198622int

120591119896

119905

120579 (21198621exp (120591

119896minus 119905)) 119889119904

le 1198622120579 (2119862

1exp (119879)) (120591

119896minus 119905) = 119862

3(120591119896minus 119905) = 120593

1198961(119905)

(50)

Suppose that 120593119896119899(119905) le 120593

119896119899(119905) holds for some 119899 ge 1


120591119896]

we obtain

120593119896119899+1

(119905) = E sup119904isin[119905120591119896]

11989021205731199041003816100381610038161003816119884119899+2 (119904) minus 119884119899+1 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119903isin[119904120591119896]

11989021205731199031003816100381610038161003816119884119899+1 (119903) minus 119884119899 (119903)

10038161003816100381610038162

)119889119904

= 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904

le 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 120593

119896119899+1(119905)

(51)


120593119896119899(119905) le 120593

119896119899(119905) (52)


119896minus1 120591

119896] Note that


119896minus1 120591



the function 120593119896(119905) by 120593

119896119899(119905) darr 120593



119896minus1 120591

119896] By the


119896(119905) we get

120593119896(119905) = lim

119899rarrinfin

1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (120593119896(119904)) 119889119904

(53)


le 119905 le 120591119896 Since int

0+(119889119906120579(119906)) = infin the Bihari

inequality implies

120593119896(119905) = 0 for each 119905 isin [120591

119896minus1 120591

119896] (54)


lim119899rarrinfin

120593119896119899(119905) = 0 (55)


Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)


119899) is


F ([0 119879]119867) times

H2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)





119879-measurable


(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)


119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))


2

(O))


2

(O)) 997888rarr R

(61)


(H1) 119886119894119895 119887

119894 R119899


119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|




1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)


2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))


119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)



119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1199101015840(sdot) 119910(sdot) isin S2


F ([0 119879]L(Γ119867))


119899 119885


S2

F ([0 119879]119867) timesH2


119884119899(119905) = 119890

119860(119879minus119905)120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)

times119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885119899(119904) 119889119882 (119904)

(27)


119899(119905) 119885

119899(119905))


119899(119905) 119885

119899(119905)) 119905 isin

[0 119879]


1and 119862

2such that

(i)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905)

(ii)Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904 le 21198621exp (119879 minus 119905)

(iii)E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le 1198622int

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(28)



yields

10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

le 2120579 (100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 2120579 (1003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

)

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

le 4119886 + 2119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 21198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 2119897 (100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+1003816100381610038161003816119885119899

(119904)10038161003816100381610038162

) + 21003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

(29)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) + Eint119879

119905

11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))]

10038161003816100381610038161003816

2

119889119904

le 241198722

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+12119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904 10038161003816100381610038161003816119891 (119904 119884

1015840

119899minus1(119904)

1198851015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le 1198621+24119872

2

119860

120573E

times int

119879

119905

E1015840[119890

2120573119904[119887100381610038161003816100381610038161198841015840

119899minus1(119904)10038161003816100381610038161003816

2

+ 1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 119897100381610038161003816100381610038161198851015840

119899(119904)10038161003816100381610038161003816

2

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

]] 119889119904

= 1198621+48119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[1198871003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

+ 1198971003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

(30)

where

1198621= 24119872

2

1198601198902120573119879

E100381610038161003816100381612058510038161003816100381610038162

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[2119886 +1003816100381610038161003816119891 (119904 0 0 0 0)

10038161003816100381610038162

] 119889119904 + 1

(31)

If we set 120573 = 961198722

119860max119887 119897 we can obtain

sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) +1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119884119899minus1 (119904)

10038161003816100381610038162

] 119889119904

le 1198621+1

2int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

(32)


sup119899isinN

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899 (119904)

10038161003816100381610038162

) le 21198621exp (119879 minus 119905

2)

le 21198621exp (119879 minus 119905)

(33)




int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 21198621+ int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

le 21198621+ 2119862

1int

119879

119905

exp (119879 minus 119904) 119889119904

= 21198621exp (119879 minus 119905)

(34)


E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

]

le E1015840[120579 (

100381610038161003816100381610038161198841015840

119899(119904) minus 119884

1015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 119897100381610038161003816100381610038161198851015840

119899+1(119904) minus 119885

1015840

119899(119904)10038161003816100381610038161003816

2

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

= E [120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

(35)

By Lemma 1 we have

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

+ Eint119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))]

10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904

E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)

+ 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

] 119889119904

(36)


(1 minus24119872

2

119860119897

120573)Eint

119879

119905

11989021205731199041003817100381710038171003817119885119899+1

(119904) minus 119885119899(119904)10038171003817100381710038172

119889119904 ge 0 (37)

Then

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(38)

where we set 1198622= (24119872

2

119860120573)119890

2120573119879



119898= 119879 by setting 120591

119896= 119896120575 119896 = 1 2 3 119898 with

120575 = 119879119898


120591119896] define

1198623= 119862

2120579 (2119862

1exp (119879))

1205931198961(119905) = 119862

3(120591119896minus 119905)

120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 119899 ge 1

(39)


0 le 120593119896119899(119905) le 120593


1198961(119905) (40)


120591119896]


1205931198962(119905) = 119862

2int

120591119896

119905

120579 (1205931198961(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (1198623(120591119896minus 119904)) 119889119904

le 1198623(120591119896minus 119905) = 120593

1198961(119905)

(41)


1198622120579 (119862

3(120591119896minus 119905)) le 119862

3= 119862

2120579 (2119862

1exp (119879)) (42)


or

1198623(120591119896minus 119905) = 119862

2120579 (2119862

1exp (119879)) (120591

119896minus 119905) le 2119862

1exp (119879)

(43)


if

1198622(119886 + 119887) (120591

119896minus 119905) le 1 (44)


120591119896] 119896 = 1 2 119898 if 120591

119896minus

120591119896minus1





120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 le 119862

2int

120591119896

119905

120579 (120593119896119899minus1

(119904)) 119889119904

= 120593119896119899(119905) forall119905 isin [120591

119896minus1 120591

119896]

(45)






E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) + Eint119879

119905

119890212057311990410038161003816100381610038161003816

119885 (119904) minus 119885 (119904)10038161003816100381610038161003816

2

119889119904

le24119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

) + 11989710038161003816100381610038161003816119885 (119904) minus 119885 (119904)

10038161003816100381610038161003816

2

] 119889119904

(46)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

)] 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

119890212057311990310038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)119889119904

(47)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) = 0 (48)



119899) defined by (27)

satisfies

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

997888rarr 0 forall0 le 119905 le 119879 (49)


119896minus1 120591

119896] we set 120593

119896119899(119905) =

E sup119904isin[119905120591119896]

1198902120573119904|119884119899+1

(119904) minus 119884119899(119904)|

2 By Lemmas 6 and 7

1205931198961(119905) = E sup

119904isin[119905120591119896]

119890212057311990410038161003816100381610038161198842 (119904) minus 1198841 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119904le119903le120591119896

119890212057311990310038161003816100381610038161198841 (119903) minus 1198840 (119903)

10038161003816100381610038162

)119889119904

le 1198622int

120591119896

119905

120579 (21198621exp (120591

119896minus 119905)) 119889119904

le 1198622120579 (2119862

1exp (119879)) (120591

119896minus 119905) = 119862

3(120591119896minus 119905) = 120593

1198961(119905)

(50)

Suppose that 120593119896119899(119905) le 120593

119896119899(119905) holds for some 119899 ge 1


120591119896]

we obtain

120593119896119899+1

(119905) = E sup119904isin[119905120591119896]

11989021205731199041003816100381610038161003816119884119899+2 (119904) minus 119884119899+1 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119903isin[119904120591119896]

11989021205731199031003816100381610038161003816119884119899+1 (119903) minus 119884119899 (119903)

10038161003816100381610038162

)119889119904

= 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904

le 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 120593

119896119899+1(119905)

(51)


120593119896119899(119905) le 120593

119896119899(119905) (52)


119896minus1 120591

119896] Note that


119896minus1 120591



the function 120593119896(119905) by 120593

119896119899(119905) darr 120593



119896minus1 120591

119896] By the


119896(119905) we get

120593119896(119905) = lim

119899rarrinfin

1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (120593119896(119904)) 119889119904

(53)


le 119905 le 120591119896 Since int

0+(119889119906120579(119906)) = infin the Bihari

inequality implies

120593119896(119905) = 0 for each 119905 isin [120591

119896minus1 120591

119896] (54)


lim119899rarrinfin

120593119896119899(119905) = 0 (55)


Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)


119899) is


F ([0 119879]119867) times

H2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)





119879-measurable


(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)


119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))


2

(O))


2

(O)) 997888rarr R

(61)


(H1) 119886119894119895 119887

119894 R119899


119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|




1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)


2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))


119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)



119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int

119879

119905

sup119899isinN

E [11989021205731199041003816100381610038161003816119885119899

(119904)10038161003816100381610038162

] 119889119904

le 21198621+ int

119879

119905

sup119899isinN

E[ sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899minus1 (119903)

10038161003816100381610038162

]119889119904

le 21198621+ 2119862

1int

119879

119905

exp (119879 minus 119904) 119889119904

= 21198621exp (119879 minus 119905)

(34)


E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904) 119884

119899minus1(119904) 119885

119899(119904))

10038161003816100381610038161003816

2

]

le E1015840[120579 (

100381610038161003816100381610038161198841015840

119899(119904) minus 119884

1015840

119899minus1(119904)10038161003816100381610038161003816

2

) + 119897100381610038161003816100381610038161198851015840

119899+1(119904) minus 119885

1015840

119899(119904)10038161003816100381610038161003816

2

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

= E [120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

]

+ 120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) + 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

(35)

By Lemma 1 we have

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

+ Eint119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904 10038161003816100381610038161003816

E1015840[119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))]

10038161003816100381610038161003816

2

119889119904

le12119872

2

119860

120573E

times int

119879

119905

1198902120573119904

E1015840[10038161003816100381610038161003816119891 (119904 119884

1015840

119899(119904) 119885

1015840

119899+1(119904) 119884

119899(119904) 119885

119899+1(119904))

minus 119891 (119904 1198841015840

119899minus1(119904) 119885

1015840

119899(119904)

119884119899minus1

(119904) 119885119899(119904))

10038161003816100381610038161003816

2

] 119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)

+ 1198971003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

] 119889119904

(36)


(1 minus24119872

2

119860119897

120573)Eint

119879

119905

11989021205731199041003817100381710038171003817119885119899+1

(119904) minus 119885119899(119904)10038171003817100381710038172

119889119904 ge 0 (37)

Then

E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

) 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

11989021205731199031003816100381610038161003816119884119899 (119903) minus 119884119899minus1 (119903)

10038161003816100381610038162

)119889119904

(38)

where we set 1198622= (24119872

2

119860120573)119890

2120573119879



119898= 119879 by setting 120591

119896= 119896120575 119896 = 1 2 3 119898 with

120575 = 119879119898


120591119896] define

1198623= 119862

2120579 (2119862

1exp (119879))

1205931198961(119905) = 119862

3(120591119896minus 119905)

120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 119899 ge 1

(39)


0 le 120593119896119899(119905) le 120593


1198961(119905) (40)


120591119896]


1205931198962(119905) = 119862

2int

120591119896

119905

120579 (1205931198961(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (1198623(120591119896minus 119904)) 119889119904

le 1198623(120591119896minus 119905) = 120593

1198961(119905)

(41)


1198622120579 (119862

3(120591119896minus 119905)) le 119862

3= 119862

2120579 (2119862

1exp (119879)) (42)


or

1198623(120591119896minus 119905) = 119862

2120579 (2119862

1exp (119879)) (120591

119896minus 119905) le 2119862

1exp (119879)

(43)


if

1198622(119886 + 119887) (120591

119896minus 119905) le 1 (44)


120591119896] 119896 = 1 2 119898 if 120591

119896minus

120591119896minus1





120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 le 119862

2int

120591119896

119905

120579 (120593119896119899minus1

(119904)) 119889119904

= 120593119896119899(119905) forall119905 isin [120591

119896minus1 120591

119896]

(45)






E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) + Eint119879

119905

119890212057311990410038161003816100381610038161003816

119885 (119904) minus 119885 (119904)10038161003816100381610038161003816

2

119889119904

le24119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

) + 11989710038161003816100381610038161003816119885 (119904) minus 119885 (119904)

10038161003816100381610038161003816

2

] 119889119904

(46)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

)] 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

119890212057311990310038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)119889119904

(47)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) = 0 (48)



119899) defined by (27)

satisfies

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

997888rarr 0 forall0 le 119905 le 119879 (49)


119896minus1 120591

119896] we set 120593

119896119899(119905) =

E sup119904isin[119905120591119896]

1198902120573119904|119884119899+1

(119904) minus 119884119899(119904)|

2 By Lemmas 6 and 7

1205931198961(119905) = E sup

119904isin[119905120591119896]

119890212057311990410038161003816100381610038161198842 (119904) minus 1198841 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119904le119903le120591119896

119890212057311990310038161003816100381610038161198841 (119903) minus 1198840 (119903)

10038161003816100381610038162

)119889119904

le 1198622int

120591119896

119905

120579 (21198621exp (120591

119896minus 119905)) 119889119904

le 1198622120579 (2119862

1exp (119879)) (120591

119896minus 119905) = 119862

3(120591119896minus 119905) = 120593

1198961(119905)

(50)

Suppose that 120593119896119899(119905) le 120593

119896119899(119905) holds for some 119899 ge 1


120591119896]

we obtain

120593119896119899+1

(119905) = E sup119904isin[119905120591119896]

11989021205731199041003816100381610038161003816119884119899+2 (119904) minus 119884119899+1 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119903isin[119904120591119896]

11989021205731199031003816100381610038161003816119884119899+1 (119903) minus 119884119899 (119903)

10038161003816100381610038162

)119889119904

= 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904

le 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 120593

119896119899+1(119905)

(51)


120593119896119899(119905) le 120593

119896119899(119905) (52)


119896minus1 120591

119896] Note that


119896minus1 120591



the function 120593119896(119905) by 120593

119896119899(119905) darr 120593



119896minus1 120591

119896] By the


119896(119905) we get

120593119896(119905) = lim

119899rarrinfin

1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (120593119896(119904)) 119889119904

(53)


le 119905 le 120591119896 Since int

0+(119889119906120579(119906)) = infin the Bihari

inequality implies

120593119896(119905) = 0 for each 119905 isin [120591

119896minus1 120591

119896] (54)


lim119899rarrinfin

120593119896119899(119905) = 0 (55)


Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)


119899) is


F ([0 119879]119867) times

H2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)





119879-measurable


(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)


119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))


2

(O))


2

(O)) 997888rarr R

(61)


(H1) 119886119894119895 119887

119894 R119899


119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|




1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)


2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))


119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)



119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










or

1198623(120591119896minus 119905) = 119862

2120579 (2119862

1exp (119879)) (120591

119896minus 119905) le 2119862

1exp (119879)

(43)


if

1198622(119886 + 119887) (120591

119896minus 119905) le 1 (44)


120591119896] 119896 = 1 2 119898 if 120591

119896minus

120591119896minus1





120593119896119899+1

(119905) = 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 le 119862

2int

120591119896

119905

120579 (120593119896119899minus1

(119904)) 119889119904

= 120593119896119899(119905) forall119905 isin [120591

119896minus1 120591

119896]

(45)






E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) + Eint119879

119905

119890212057311990410038161003816100381610038161003816

119885 (119904) minus 119885 (119904)10038161003816100381610038161003816

2

119889119904

le24119872

2

119860

120573E

times int

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

) + 11989710038161003816100381610038161003816119885 (119904) minus 119885 (119904)

10038161003816100381610038161003816

2

] 119889119904

(46)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (10038161003816100381610038161003816119884 (119904) minus (119904)

10038161003816100381610038161003816

2

)] 119889119904

le 1198622Eint

119879

119905

120579(E sup119904le119903le119879

119890212057311990310038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

)119889119904

(47)


E( sup119905le119904le119879

119890212057311990410038161003816100381610038161003816

119884 (119904) minus (119904)10038161003816100381610038161003816

2

) = 0 (48)



119899) defined by (27)

satisfies

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

997888rarr 0 forall0 le 119905 le 119879 (49)


119896minus1 120591

119896] we set 120593

119896119899(119905) =

E sup119904isin[119905120591119896]

1198902120573119904|119884119899+1

(119904) minus 119884119899(119904)|

2 By Lemmas 6 and 7

1205931198961(119905) = E sup

119904isin[119905120591119896]

119890212057311990410038161003816100381610038161198842 (119904) minus 1198841 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119904le119903le120591119896

119890212057311990310038161003816100381610038161198841 (119903) minus 1198840 (119903)

10038161003816100381610038162

)119889119904

le 1198622int

120591119896

119905

120579 (21198621exp (120591

119896minus 119905)) 119889119904

le 1198622120579 (2119862

1exp (119879)) (120591

119896minus 119905) = 119862

3(120591119896minus 119905) = 120593

1198961(119905)

(50)

Suppose that 120593119896119899(119905) le 120593

119896119899(119905) holds for some 119899 ge 1


120591119896]

we obtain

120593119896119899+1

(119905) = E sup119904isin[119905120591119896]

11989021205731199041003816100381610038161003816119884119899+2 (119904) minus 119884119899+1 (119904)

10038161003816100381610038162

le 1198622int

120591119896

119905

120579(E sup119903isin[119904120591119896]

11989021205731199031003816100381610038161003816119884119899+1 (119903) minus 119884119899 (119903)

10038161003816100381610038162

)119889119904

= 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904

le 1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 120593

119896119899+1(119905)

(51)


120593119896119899(119905) le 120593

119896119899(119905) (52)


119896minus1 120591

119896] Note that


119896minus1 120591



the function 120593119896(119905) by 120593

119896119899(119905) darr 120593



119896minus1 120591

119896] By the


119896(119905) we get

120593119896(119905) = lim

119899rarrinfin

1198622int

120591119896

119905

120579 (120593119896119899(119904)) 119889119904 = 119862

2int

120591119896

119905

120579 (120593119896(119904)) 119889119904

(53)


le 119905 le 120591119896 Since int

0+(119889119906120579(119906)) = infin the Bihari

inequality implies

120593119896(119905) = 0 for each 119905 isin [120591

119896minus1 120591

119896] (54)


lim119899rarrinfin

120593119896119899(119905) = 0 (55)


Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)


119899) is


F ([0 119879]119867) times

H2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)





119879-measurable


(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)


119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))


2

(O))


2

(O)) 997888rarr R

(61)


(H1) 119886119894119895 119887

119894 R119899


119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|




1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)


2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))


119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)



119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Then

E sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

le max1le119896le119898

E sup119904isin[120591119896minus1120591119896]

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

= max1le119896le119898

120593119896119899(119905) 997888rarr 0

(56)


E( sup119905le119904le119879

11989021205731199041003816100381610038161003816119884119899+1 (119904) minus 119884119899 (119904)

10038161003816100381610038162

) + (1 minus24119872

2

119860119897

120573)E

times int

119879

119905

11989021205731199041003816100381610038161003816119885119899+1

(119904) minus 119885119899(119904)10038161003816100381610038162

119889119904

le24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[120579 (1003816100381610038161003816119884119899 (119904) minus 119884119899minus1 (119904)

10038161003816100381610038162

)] 119889119904

(57)


119899) is


F ([0 119879]119867) times

H2


119884 (119905) = 119890119860(119879minus119905)

120585

+ int

119879

119905

E1015840[119890

119860(119904minus119905)119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119885 (119904) 119889119882 (119904)

(58)





119879-measurable


(R) of the form

119871 =

119899

sum

119894119895=1

119886119894119895(119909)

1205972

120597119909119894120597119909

119895

+

119899

sum

119894=1

119887119894(119909)

120597

120597119909119894

(59)


119889119884 (119905 119909)

= (119871119884 (119905 119909) + E1015840

times [119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))]) 119889119905

+ 119885 (119905 119909) 119889119861 (119905) ae on (0 119879) times O

119884 (119905 119909) = 0 ae on (0 119879) times 120597O

119884 (119879 119909) = 120585 (119879 119909) ae onO(60)

where

119884 [0 119879] times O 997888rarr R

119885 [0 119879] times O 997888rarr L (R119899 119871

2

(O))


2

(O))


2

(O)) 997888rarr R

(61)


(H1) 119886119894119895 119887

119894 R119899


119894119895=1119886119894119895(119909)119908

119894119908119895ge 120582|119908|




1003816100381610038161003816

le 119862 [10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161 minus 2

1003816100381610038161003816 +10038161003816100381610038161199101 minus 1199102

1003816100381610038161003816 +10038161003816100381610038161199111 minus 1199112

1003816100381610038161003816]

(62)


2 119910

1 119910

2isin R

1

2 119911

1 119911

2isin

L(R119899 119871

2(O))



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))


119860119884 (119905 sdot) = 119871119884 (119905 sdot) (63)



119891 (119905 1198841015840

(119905) 1198851015840

(119905) 119884 (119905) 119885 (119905)) (119909)

= 119892 (119905 119909 1198841015840

(119905 119909) 1198851015840

(119905 119909) 119884 (119905 119909) 119885 (119905 119909))

(64)



1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198712(0 119879 119871

2(Ω 119871

2(O))) times 1198712F (0 119879 119871

2(R119899

1198712(Ω 119871

2(O))))



119889119883 (119904) = 119861119883 (119904) 119889119904 + E1015840[119887 (119904 119883

1015840

(119904) 119883 (119904))] 119889119904

+ E1015840[120590 (119904 119883

1015840

(119904) 119883 (119904))] 119889119882 (119904)

119883 (119905) = 119909 isin 119870

(65)



sup119905isin[0119879]

|119890119905119861|



119883 (119904) = 119890119861(119904minus119905)

119909

+ int

119904

119905

119890119861(119904minus120591)

E1015840[119887 (120591 119883

1015840

(120591) 119883 (120591))] 119889120591

+ int

119904

119905

119890119861(119904minus120591)

E1015840[120590 (120591119883

1015840

(120591) 119883 (120591))] 119889119882 (120591)

119904 isin [119905 119879]

(66)


which satisfies10038161003816100381610038161003816119887 (119905 119909

1015840 119909) minus 119887 (119905 119910

1015840 119910)

10038161003816100381610038161003816

2

le 1198711(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816

2

+1003816100381610038161003816119909 minus 119910

10038161003816100381610038162

)

119905 isin [0 119879] 1199091015840 119909 119910

1015840 119910 isin 119870

(67)




1015840 119909) isin L(Γ 119870) and

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2119904minus120574(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

10038161003816100381610038161003816119890119904119861120590 (119905 119909

1015840 119909) minus 119890

119904119861120590 (119905 119910

1015840 119910)

10038161003816100381610038161003816L(Γ119870)

le 1198712119904minus120574(100381610038161003816100381610038161199091015840minus 119910

101584010038161003816100381610038161003816+1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

10038161003816100381610038161003816120590 (119905 119909

1015840 119909)

10038161003816100381610038161003816L(Γ119870)le 119871

2(1 +

10038161003816100381610038161003816119909101584010038161003816100381610038161003816+ |119909|)

(68)



F ([119905 119879] 119870)






1and119871


Also assume that

Eint119879

0

10038161003816100381610038161003816E1015840[119887120572(119904 119883

1015840

(119904) 119883 (119904))]

minus E1015840[1198871205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

Eint119879

0

10038161003816100381610038161003816E1015840[120590

120572(119904 119883

1015840

(119904) 119883 (119904))]

minusE1015840[120590

1205720(119904 119883

1015840

(119904) 119883 (119904))]10038161003816100381610038161003816

2

119889119904 997888rarr 0

(69)


F ([0 119879] 119870)If we denote by 119883120572


120572 120590



sup119904isin[0119879]

E1003816100381610038161003816119883

120572

(119904) minus 1198831205720 (119904)

10038161003816100381610038162

997888rarr 0 119886119904 120572 997888rarr 1205720 (70)



U = V (sdot) isin 1198712F(0 119879 119880)

| V119905(120596

1015840 120596) [0 119879] times Ω times Ω


(71)





119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119889119884119905(119909) = minus119860119884

119905(119909) 119889119905

minus E1015840[119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) V

119905)] 119889119905

+ 119885119905(119909) 119889119882 (119905) 119905 isin [0 119879]

119884119879(119909) = 120585 (119909) 119909 isin O

(72)




119869 (V) = Eint119879

0

intO

E1015840[ℎ (119904 119909 (119884

119904(119909))

1015840

(119885119904(119909))

1015840

119884119904(119909) 119885

119904(119909) V

119904)] 119889119909 119889119904

+E1015840intO

119892 (119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

(73)

where



(74)


EintO

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909 = 0 (75)

where



119869 (119906) = minVisinUad

119869 (V) (77)


in force


119905119860119891

1198712(O) le 119872

119860119891


119860




100381610038161003816100381610038161198911199101015840

10038161003816100381610038161003816+10038161003816100381610038161198911199111015840

1003816100381610038161003816 +10038161003816100381610038161003816119891119910

10038161003816100381610038161003816+1003816100381610038161003816119891119911

1003816100381610038161003816 +1003816100381610038161003816119891V

1003816100381610038161003816 +10038161003816100381610038161003816Φ1199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816Φ119910

10038161003816100381610038161003816le 119862

10038161003816100381610038161003816ℎ1199101015840

10038161003816100381610038161003816+1003816100381610038161003816ℎ1199111015840

1003816100381610038161003816 +10038161003816100381610038161003816ℎ119910

10038161003816100381610038161003816+1003816100381610038161003816ℎ119911

1003816100381610038161003816 +100381610038161003816100381610038161198921199101015840

10038161003816100381610038161003816+10038161003816100381610038161003816119892119910

10038161003816100381610038161003816

le 119862 (1 +10038161003816100381610038161003816119910101584010038161003816100381610038161003816+10038161003816100381610038161003816119911101584010038161003816100381610038161003816+10038161003816100381610038161199101003816100381610038161003816 + |119911| + |V|)

(78)




(119860119891) (119909) =

119899

sum

119894119895=1

119886119894119895(119909)

1205972119891

120597119909119894120597119909

119895

(119909) +

119899

sum

119894=1

119887119894(119909)

120597119891

120597119909119894

(119909) (79)


119894satisfy

condition (H1)


119906120576

119905=

V119905 119905 isin [119903 119903 + 120576]

119906119905 119905 isin [0 119879] [119904 119904 + 120576]

(80)




119891 (119905) = 119891 (119905 119909 (119884119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

119905)

119891 (119906120576

119905) = 119891 (119905 119909 (119884

119905(119909))

1015840

(119885119905(119909))

1015840

119884119905(119909) 119885

119905(119909) 119906

120576

119905)

(81)


119889119870120576

119905(119909) = minus119860119870

120576

119905(119909) 119889119905

minus E1015840[119891

1199101015840 (119905) (119870

120576

119905(119909))

1015840

+ 119891119910(119905) 119870

120576

119905(119909)

+ 1198911199111015840 (119905) (119876

120576

119905(119909))

1015840

+ 119891119911(119905) 119876

120576

119905(119909)

+1

120576(119891 (119906

120576

119905) minus 119891 (119905))] 119889119905 + 119876

120576

119905(119909) 119889119882 (119905)

119870120576

119879(119909) = 0

(82)


E[ sup119905isin[0119879]

1003816100381610038161003816119870120576

119905

10038161003816100381610038162

+ int

119879

0

1003816100381610038161003816119876120576

119905

10038161003816100381610038162

119889119905] lt infin (83)




lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











lim120576rarr0

E[ sup119904isin[119905119879]

1003816100381610038161003816100381610038161003816100381610038161003816

119884120576

119904minus 119884

119904

120576minus 119870

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

1003816100381610038161003816100381610038161003816100381610038161003816

119885120576

119904minus 119885

119904

120576minus 119876

120576

119904

1003816100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(84)

Proof We define

120578120576

119904=119884120576

119904minus 119884

119904

120576minus 119870

120576

119904 120577

120576

119904=119885120576

119904minus 119885

119904

120576minus 119876

120576

119904 119904 isin [0 119879]

(85)


Λ120576

119904= ((119884

120576

119904)1015840

(119885120576

119904)1015840

119884120576

119904 119885

120576

119904)

119891 (119904 120582) = 119891 (119904 1198841015840

119904+ 120582(119884

120576

119904minus 119884

119904)1015840

1198851015840

119904+ 120582(119885

120576

119904minus 119885

119904)1015840

119884119904+ 120582 (119884

120576

119904minus 119884

119904)

119885119904+ 120582 (119885

120576

119904minus 119885

119904) 119906

120576

119904)

(86)


119904 119885

120576

119904) (119884

119904 119885

119904) and (119870120576

119904 119876

120576

119904) (120578120576

119904 120577

120576

119904)


119889120578120576

119904= minus119860120578

120576

119904119889119904 minus E

1015840

[119871 (119904 120576)] 119889119904 + 120577120576

119904119889119882 (119904)

120578120576

119879= 0

(87)

with

119871 (119904 120576) =1

120576(119891 (119904 Λ

120576

119904 119906

120576

119904) minus 119891 (119906

120576

119904))

minus 1198911199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= ((120578120576

119904)1015840

+ (119870120576

119904)1015840

) int

1

0

1198911199101015840 (119904 120582) 119889120582 + (120578

120576

119904+ 119870

120576

119904)

times int

1

0

119891119910(119904 120582) 119889120582 minus 119891

1199101015840 (119904) (119870

120576

119904)1015840

minus 119891119910(119904) 119870

120576

119904

+ ((120577120576

119904)1015840

+ (119876120576

119904)1015840

)int

1

0

1198911199111015840 (119904 120582) 119889120582 + (120577

120576

119904+ 119876

120576

119904)

times int

1

0

119891119911(119904 120582) 119889120582 minus 119891

1199111015840 (119904) (119876

120576

119904)1015840

minus 119891119911(119904) 119876

120576

119904

= (120578120576

119904)1015840

int

1

0

1198911199101015840 (119904 120582) 119889120582 + 120578

120576

119904int

1

0

119891119910(119904 120582) 119889120582

+ (120577120576

119904)1015840

int

1

0

1198911199111015840 (119904 120582) 119889120582 + 120577

120576

119904int

1

0

119891119911(119904 120582) 119889120582 + 120574

120576

119904

(88)

where we denote

120574120576

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

+ 119870120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

+ (119876120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

+ 119876120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(89)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ Eint119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le12119872

2

119860

120573int

119879

119905

1198902120573119904

E [10038161003816100381610038161003816E1015840

[119871 (119904 120576)]10038161003816100381610038161003816

2

] 119889119904

le12119872

2

119860

120573Eint

119879

119905

1198902120573119904

E1015840[|119871 (119904 120576)|

2] 119889119904

(90)


E [E1015840[|119871 (119904 120576)|

2]]

= E [E1015840[1003816100381610038161003816119871 (119904 120576) minus 120574

120576

119904+ 120574

120576

119904

10038161003816100381610038162

]]

le 81198622E [E

1015840[10038161003816100381610038161003816(120578

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+10038161003816100381610038161003816(120577

120576

119904)101584010038161003816100381610038161003816

2

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

]]

+ 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

le 161198622E [

1003816100381610038161003816120578120576

119904

10038161003816100381610038162

+1003816100381610038161003816120577120576

119904

10038161003816100381610038162

] + 2E [E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]]

(91)


E sup119905le119904le119879

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

+ (1 minus192119872

2

1198601198622

120573)Eint

119879

119905

11989021205731199041003816100381610038161003816120577

120576

119904

10038161003816100381610038162

119889119904

le192119872

2

1198601198622

120573Eint

119879

119905

11989021205731199041003816100381610038161003816120578

120576

119904

10038161003816100381610038162

119889119904

+24119872

2

119860

120573Eint

119879

119905

1198902120573119904

[E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

]] 119889119904

(92)

We claim that

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904 997888rarr 0 as 120576 997888rarr 0 (93)

From (89)

120574120576

119904= 119868

1

119904+ 119868

2

119904+ 119868

3

119904+ 119868

4

119904 (94)


where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

1198681

119904= (119870

120576

119904)1015840

int

1

0

(1198911199101015840 (119904 120582) minus 119891

1199101015840 (119904)) 119889120582

1198682

119904= 119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582

1198683

119904= (119876

120576

119904)1015840

int

1

0

(1198911199111015840 (119904 120582) minus 119891

1199111015840 (119904)) 119889120582

1198684

119904= 119876

120576

119904int

1

0

(119891119911(119904 120582) minus 119891

119911(119904)) 119889120582

(95)

Then

Eint119879

119905

E1015840[1003816100381610038161003816120574120576

119904

10038161003816100381610038162

] 119889119904

le 4Eint119879

119905

E1015840[100381610038161003816100381610038161198681

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198683

119904

10038161003816100381610038161003816

2

+100381610038161003816100381610038161198684

119904

10038161003816100381610038161003816

2

] 119889119904

(96)

Take Eint119879119905E1015840[|1198682

119904|2


Eint119879

119905

E1015840[100381610038161003816100381610038161198682

119904

10038161003816100381610038161003816

2

] 119889119904

= Eint119879

119905

E1015840[(119870

120576

119904int

1

0

(119891119910(119904 120582) minus 119891

119910(119904)) 119889120582)

2

]119889119904

le Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

+ 2Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

(97)

Note that

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

= Eint119903+120576

119903

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582] 119889119904

le sup119904isin[119905119879]

E [E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119906

120576

119905) minus 119891

119910(119904)10038161003816100381610038161003816

2

119889120582]] 120576

997888rarr 0 as 120576 997888rarr 0

(98)


120576

119904|2

int1

0|119891119910(119906

120576

119905) minus 119891

119910(119904)|

2


119910(119906

120576

119905) minus 119891

119910(119904) Meanwhile 119870120576

119904is



Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

1198641015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

(99)

where (119884120576

119904 119885

120576


119889119884120576

119905= minus 119860119884

120576

119905119889119905

minus E1015840[119891 (119905 (119884

120576

119905)1015840

(119885120576

119905)1015840

119884120576

119905 119885

120576

119905 119906

120576

119905)] 119889119905

+ 119885120576

119905119889119882 (119905) 119905 isin [0 119879]

119884120576

119879= 120585

(100)

and (119884119904 119885


119889119884119905= minus 119860119884

119905119889119905

minus E1015840[119891 (119905 (119884

119905)1015840

(119885119905)1015840

119884119905 119885

119905 119906

119905)] 119889119905

+ 119885119905119889119882 (119905) 119905 isin [0 119879]

119884119879= 120585

(101)


E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906120576

119905)]

997888rarr E1015840[119891 (119904 119884

1015840

(119904) 1198851015840

(119904) 119884 (119904) 119885 (119904) 119906119905)]

(102)


(119884120576

119904 119885

120576

119904) 997888rarr (119884

119904 119885

119904) as 120576 997888rarr 0 (103)

Then

Eint119879

119905

E1015840[1003816100381610038161003816119870

120576

119904

10038161003816100381610038162

int

1

0

10038161003816100381610038161003816119891119910(119904 120582) minus 119891

119910(119906

120576

119905)10038161003816100381610038161003816

2

119889120582] 119889119904

le 11986221205822Eint

119879

119905

1003816100381610038161003816119870120576

119904

10038161003816100381610038162

times int

1

0

E1015840[100381610038161003816100381610038161003816(119884

120576

119904minus 119884

119904)1015840100381610038161003816100381610038161003816

2

+100381610038161003816100381610038161003816(119885

120576

119904minus 119885

119904)1015840100381610038161003816100381610038161003816

2

times10038161003816100381610038161003816119884120576

119904minus 119884

119904

10038161003816100381610038161003816

2

+10038161003816100381610038161003816119885120576

119904minus 119885

119904

10038161003816100381610038161003816

2

] 119889120582 119889119904

997888rarr 0 as 120576 997888rarr 0

(104)



119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879

119905E1015840[|1198682

119904|2


estimations for 1198681119904 1198683

119904 and 1198684

119904







119865 (119906) le infVisin119878

119865 (V) + 120588 (105)


(i) 119865 (119906120588) le 119865 (119906)

(ii) 119889 (119906120588 119906) le 120588


(106)



E1003816100381610038161003816V11990510038161003816100381610038162

le E100381610038161003816100381611990611990510038161003816100381610038162

+ |V|2


119905

10038161003816100381610038162

gt 0


(107)




119869120588


+

10038161003816100381610038161003816100381610038161003816Eint

O

Φ(119909 (1198840(119909))

1015840

1198840(119909)) 119889119909

10038161003816100381610038161003816100381610038161003816

2

12





119869120588

(V (sdot)) (109)



119869120588

(V (sdot)) le 119869120588

(119906 (sdot)) = 120588 (110)


(i) 119869120588 (119906120588 (sdot)) le 120588

(ii) 119889 (119906120588 (sdot) 119906 (sdot)) le 120588

(iii) 119869120588 (119906120588 (sdot)) le 119869120588



119906120576120588

119905=

V119905 119905 isin [119904 119904 + 120576]

119906120588

119905 119905 isin [0 119879] [119904 119904 + 120576]

(112)


120576120588(sdot)) (resp

(119884120588(sdot) 119885



119905 119876

120576120588

119905) is the

mild solution of

119870120576120588

119905= int

119879

119905

119890119860(119904minus119905)

E1015840

times [1198911199101015840 (119904 Λ

120588

119904 119906

120588

119904) (119870

120576120588

119904)1015840

+ 119891119910(119904 Λ

120588

119904 119906

120588

119904)119870

120576120588

119904

+ 1198911199111015840 (119904 Λ

120588

119904 119906

120588

119904) (119876

120576120588

119904)1015840

+ 119891119911(119904 Λ

120588

119904 119906

120588

119904) 119876

120576120588

119904

+1

120576(119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904))] 119889119904

minus int

119879

119905

119890119860(119904minus119905)

119876120576120588

119904119889119882 (119904)

(113)


lim120576rarr0

E[ sup119904isin[119905119879]

100381610038161003816100381610038161003816100381610038161003816

119884120576120588

119904minus 119884

120588

119904

120576minus 119870

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

] = 0 forall119905 isin [0 119879]

lim120576rarr0

Eint119879

0

100381610038161003816100381610038161003816100381610038161003816

119885120576120588

119904minus 119885

120588

119904

120576minus 119876

120576120588

119904

100381610038161003816100381610038161003816100381610038161003816

2

119889119904 = 0

(114)



1

120576E [Φ ((119884

120576120588

0)1015840

119884120576120588

0) minus Φ((119884

120588

0)1015840

119884120588

0)]

= E [Φ1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ 119900 (120576)


1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

120576(119869 (119906

120576120588) minus 119869 (119906

120588))

= E [1198921199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+ 119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+ Δ120576+1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + 119900 (120576)

(115)

where

Δ120576= Eint

119879

0

E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119870

120576120588

119904)1015840

+ ℎ1199111015840 (119904 Λ

120588

119904 119906

120576120588

119904) (119876

120576120588

119904)1015840

+ ℎ119910(119904 Λ

120588

119904 119906

120576120588

119904)119870

120576120588

119904

+ ℎ119911(119904 Λ

120588

119904 119906

120576120588

119904) 119876

120576120588

119904] 119889119904

(116)



119889(119906120576120588(sdot) 119906

120588(sdot)) le 120576 we get

minus120588 le1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot))) (117)

By Lemma 17

1

120576(119869

120588(119906

120576120588

(sdot)) minus 119869120588(119906

120588

(sdot)))

=(119869

120588(119906

120576120588(sdot)))

2

minus (119869120588(119906

120588(sdot)))

2

120576 (119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot)))

=119869 (119906

120576120588(sdot)) + 119869 (119906

120588(sdot)) minus 2119869 (119906 (sdot)) + 2120588

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times119869 (119906

120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] + E [Φ ((119884

120588

0)1015840

119884120588

0)]

119869120588 (119906120576120588 (sdot)) + 119869120588 (119906120588 (sdot))

times

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

997888rarr 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

(118)

where we set

119897120588

1=119869 (119906

120588(sdot)) minus 119869 (119906 (sdot)) + 120588

119869120588 (119906120588 (sdot))

119897120588

2=

E [Φ ((119884120588

0)1015840

119884120588

0)]

119869120588 (119906120588 (sdot))

(119)

and use the limit

119869 (119906120576120588

(sdot)) 997888rarr 119869 (119906120588

(sdot))

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] 997888rarr E [Φ ((119884

120588

0)1015840

119884120588

0)]

(120)


1|2

+ |119897120588

2|2


1 119897120588


1 119897120588

2) such that

lim120588rarr0

119897120588

1 119897120588

2 = 119897

1 1198972

1003816100381610038161003816119897110038161003816100381610038162

+1003816100381610038161003816119897210038161003816100381610038162

= 1

(121)


minus120588 le 119897120588

1

119869 (119906120576120588(sdot)) minus 119869 (119906

120588(sdot))

120576

+ 119897120588

2

E [Φ ((119884120576120588

0)1015840

119884120576120588

0)] minus E [Φ ((119884

120588

0)1015840

119884120588

0)]

120576

= 119897120588

1Δ120576+ 119897

120588

1E [119892

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+119892119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0]

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904) minus ℎ (119904 Λ

120588

119904 119906

120588

119904)] 119889119904

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0)1015840

119884120588

0) (119870

120576120588

0)1015840

+Φ119910((119884

120588

0)1015840

119884120588

0)119870

120576120588

0] + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(122)


119889119875120588

(119905)

= 119860lowast119875120588

(119905) 119889119905

+ E1015840[119891

1199101015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119910(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905) + 119897120588

1ℎ1199101015840 (119905 Λ

120588

119905 119906

120588

119905)

+ 119897120588

1ℎ119910(119905 Λ

120588

119905 119906

120588

119905)] 119889119905


+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ E1015840[119891

1199111015840 (119905 Λ

120588

119905 119906

120588

119905) (119875

120588

(119905))1015840

+ 119891119911(119905 Λ

120588

119905 119906

120588

119905) 119875

120588

(119905)

+ 119897120588

1ℎ1199111015840 (119905 Λ

120588

119905 119906

120588

119905) + 119897

120588

1ℎ119911(119905 Λ

120588

119905 119906

120588

119905)] 119889119882 (119905)

119875120588

(0) = 119897120588

1E [119892

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ 119892119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

+ 119897120588

2E [Φ

1199101015840 ((119884

120588

0(119909))

1015840

119884120588

0(119909))

+ Φ119910((119884

120588

0(119909))

1015840

119884120588

0(119909))]

(123)




119875 (119905) isin S2

F ([0 119879] 119870) (124)


119875 (119905) = 1198971E [119892

1199101015840 (119884

1015840

0 119884

0) + 119892

119910(119884

1015840

0 119884

0)]

+ 1198972E [Φ

1199101015840 (119884

1015840

0 119884

0) + Φ

119910(119884

1015840

0 119884

0)]

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199101015840 (119904) 119875

1015840

(119904) + 119891119910(119904) 119875 (119904)

+ 1198971ℎ1199101015840 (119904) + 119897

1ℎ119910(119904)] 119889119904

+ int

119905

0

119890119860lowast(119905minus119904)

E1015840[119891

1199111015840 (119904) 119875

1015840

(119904) + 119891119911(119904) 119875 (119904)

+ 1198971ℎ1199111015840 (119904) + 119897

1ℎ119911(119904)] 119889119882 (119904)

(125)


119905


E [119875120588

(0)119870120576120588

0]

= Eint119879

0

1

120576119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904) minus 119891 (119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119870120576120588

119904E1015840[ℎ

1199101015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119910(119904 Λ

120588

119904 119906

120588

119904)]

minus 119897120588

1119876120576120588

119904E1015840[ℎ

1199111015840 (119904 Λ

120588

119904 119906

120588

119904) + ℎ

119911(119904 Λ

120588

119904 119906

120588

119904)] 119889119904

(126)




H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 V

119905 119875 (119905))

ge H (119905 1198841015840

119905 119885

1015840

119905 119884

119905 119885

119905 119906

119905 119875 (119905))

119886119890 119886119904 forallV isin U119886119889

(127)



H (119905 119910 119910 119911 V 119901) = 1198971ℎ (119905 119910 119910 119911 V)

+ 119901119891 (119905 119910 119910 119911 V) (128)


minus120588 le1

120576Eint

119879

0

119875120588

(119904)E1015840[119891 (119904 Λ

120588

119904 119906

120576120588

119904)

minus 119891 (119904 Λ120588

119904 119906

120588

119904)] 119889119904

+119897120588

1

120576Eint

119879

0

E1015840[ℎ (119904 Λ

120588

119904 119906

120576120588

119904)

minus ℎ (119904 Λ120588

119904 119906

120588

119904)] 119889119904 + (119897

120588

1+ 119897

120588

2) 119900 (120576)

(129)


minus120588 le 119897120588

1E1015840[ℎ (120591 Λ

120588

120591 V

120591) minus ℎ (120591 Λ

120588

120591 119906

120588

120591)]

+ 119875120588

(119904)E1015840[119891 (120591 Λ

120588

120591 V

120591) minus 119891 (120591 Λ

120588

120591 119906

120588

120591)]

(130)


result



1= 1




119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119869 (119906) =1

2E [(119884 (0))

2] +

1

2Eint

119879

0

119873V2 (119905) 119889119905 997888rarr min

(131)

subject to

119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)]

+119862V (119905) + 119863119885 (119905) + 119863E [119885 (119905)] 119889119905

+ 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(132)



119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(133)



1

2119873V2 (119905) + 119875 (119905) 119862V (119905) ge

1

2119873119906

2

(119905) + 119875 (119905) 119862119906 (119905) (134)


119906 (119905) = minus119862

119873119875 (119905) (135)



119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585 119905 isin [0 119879]

(136)


119889119884 (119905) = minus119860119884 (119905) 119889119905

minus 119861119884 (119905) + 119861E [119884 (119905)] minus1198622

119873119875 (119905)

+ 119863119885 (119905) + 119863E [119885 (119905)] 119889119905 + 119885 (119905) 119889119882 (119905)

119884 (119879) = 120585

119889119875 (119905) = 119860lowast119875 (119905) 119889119905 + 119861119875 (119905) + 119861E [119875 (119905)] 119889119905

+ 119863119875 (119905) + 119863E [119875 (119905)] 119889119882 (119905)

119875 (0) = 119884 (0)

(137)


119875 (119905) = 120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)] (138)



2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]

= minus120593 (119905) 119861119884 (119905) minus1198622

119873120593 (119905) 119884 (119905) + 119863119885 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus (119863120601 (119905) + 119863120601 (119905) + 119863120593 (119905))E [119885 (119905)]

120593 (119905) 119885 (119905) = 119863120593 (119905) 119884 (119905)

+ (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(139)


2120593 (119905) 119860119884 (119905) + 2120601 (119905) 119860E [119884 (119905)] + 119861120593 (119905) 119884 (119905)

+ 2 (119861120601 (119905) + 119861120593 (119905) + 119861120601 (119905))E [119884 (119905)]


= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= 120593 (119905) minus119861 +1198622

119873120593 (119905) minus 119863

2119884 (119905)

+ 119884 (119905)119889120593 (119905)

119889119905+ E [119884 (119905)]

119889120601 (119905)

119889119905

+ (1206012

(119905)1198622

119873+ 2

1198622

119873120593 (119905) 120601 (119905))E [119884 (119905)]

minus1

120593 (119905)(119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))

2

E [119884 (119905)]

minus 2119863 (119863120601 (119905) + 119863120593 (119905) + 119863120601 (119905))E [119884 (119905)]

(140)


119889

119889119905120593 (119905) = 2119860

lowast120593 (119905) + 2119861120593 (119905) + 119863

2120593 (119905) minus

1198622

1198731205932

(119905)

120593 (0) = 1

(141)

119889

119889119905120601 (119905) = 2119860

lowast120601 (119905) + (

(119863 + 119863)2

120593 (119905)minus1198622

119873)120601

2

(119905)

+ 2119877 (119905) 120601 (119905) + (1198632+ 2119861 + 2119863119863)120593 (119905)

120601 (0) = 0

(142)

where 119877(119905) = 119861 + 119861 + (119863 + 119863)2

minus (1198622119873)120593(119905)


120593 (119905) = (119890minus(2119860lowast+2119861+119863

2)119905(1 minus

1198622

119873(2119860lowast + 2119861 + 1198632))

+1198622

119873(2119860lowast + 2119861 + 1198632))

minus1

(143)




119906 (119905) = minus119862

119873(120593 (119905) 119884 (119905) + 120601 (119905)E [119884 (119905)]) (144)




Acknowledgment


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Mean-Field Backward Stochastic Evolution …downloads.hindawi.com/journals/mpe/2014/718948.pdf · 2019-07-31 · 3. Mean-Field Backward Stochastic Evolution Equations