Asymptotics of the Probability Minimizing a Down-side Risk

8/9/2019 Asymptotics of the Probability Minimizing a Down-side Risk

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arXiv:1001

.2131v1

[math.PR]13Jan2010

The Annals of Applied Probability

2010, Vol. 20, No. 1, 5289DOI: 10.1214/09-AAP618

c Institute of Mathematical Statistics, 2010

ASYMPTOTICS OF THE PROBABILITY MINIMIZING

A DOWN-SIDE RISK

By Hiroaki Hata, Hideo Nagai1 and Shuenn-Jyi Sheu2

Academia Sinica, Osaka University and Academia Sinica

We consider a long-term optimal investment problem where aninvestor tries to minimize the probability of falling below a targetgrowth rate. From a mathematical viewpoint, this is a large devi-ation control problem. This problem will be shown to relate to arisk-sensitive stochastic control problem for a sufficiently large time

horizon. Indeed, in our theorem we state a duality in the relationbetween the above two problems. Furthermore, under a multidimen-sional linear Gaussian model we obtain explicit solutions for the pri-mal problem.

1. Introduction. In recent studies of finance, it has been of great concernto consider problems from risk management. In this paper, we consider theproblem of the control of a down-side risk probability for an investor. Tominimize such probabilities and obtain an optimal (or nearly optimal) port-folio, we relate the problem of the portfolio optimization with risk sensitivecriterion. In [35, 36], from the consideration of a performance index for fundswhen compared to a benchmark, a similar problem is considered. In [30, 31],problems of maximizing the up-side chance probability are studied. Thesestudies show potential applications of risk-sensitive dynamic managementto the problem of risk management.

Portfolio optimization with risk-sensitive criterion has been considered inseveral recent works (see [24, 7, 10, 1416, 18, 19, 21, 25, 28] and [29]).The problem is to maximize the expected utility of terminal wealth with theHARA utility function being considered,

max

E

1

(XT)

,(1.1)

Received August 2007; revised September 2008.1Supported in part by Grant-in-Aid for scientific research 20340019 JSPS.2

Supported by NSC Grant 96-2119-M-001-002.AMS 2000 subject classifications. 35J60, 49L20, 60F10, 91B28, 93E20.Key words and phrases. Large deviation, long-term investment, risk-sensitive stochas-

tic control, Bellman equation.

This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in The Annals of Applied Probability,2010, Vol. 20, No. 1, 5289. This reprint differs from the original in paginationand typographic detail.

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DOWN-SIDE RISK MINIMIZATION 3

for large T where the maximization is taken for in a class of admissiblestrategies. Or more generally,

max

P

1

Tlog

XTIT

c

,(1.6)

where IT is a benchmark process. For simplicity, we take It = 1 in the fol-lowing discussion. A mathematical theory was developed in [30] and [31] forthe maximization of the up-side chance probabilities. It is shown that theprobabilities in (1.5) are related to those in (1.1) with 0 < < 1 through aduality relation. A nearly optimal portfolio for (1.5) can be obtained from anoptimal portfolio for (1.1) for a particular chosen . Some idea from large de-viation theory is implicitly used in his approach. This will be described laterin this section. Interestingly, the maximization of the probability of up-sidechance is not a conventional optimization problem and, in general, is difficultto treat. Therefore, studies in [30, 31] suggest the possibility of indirectly us-ing the theory of stochastic control in such a nonconventional optimizationproblem. See also [19] for more studies on this problem. However, Sekine[33] recently tried to use a duality approach to treat such problems. In [ 30],Pham also proposed to develop a mathematical theory for the down-siderisk probability,

min

P

1

Tlog XT k

,(1.7)

or more generally,

min

P

1

Tlog

XTIT

k

.(1.8)

The problem is to minimize (1.6) and obtain an optimal (or nearly optimal)portfolio. Here k is considered such that the event has small probabilities.Hence we are dealing with a rare event. From the consideration of finance,such rare events in down-side risk are not favorable to an investor. Thereforeits occurrence may result in a significant consequence in portfolio manage-ment. Hence the study of (1.7) or (1.8) may be of meaningful implicationsin finance. See interesting discussions in [5, 6] and [35, 36] for some relatedproblems where the consideration is the performance index for funds. In this

paper, we develop some mathematical analyses for (1.7). Similar to [30, 31],we will show a dual relation between (1.1) and (1.7) for large T. The resultsays that for k, there is a correspondence (k) < 0 such that an optimalportfolio of (1.1) with = (k) is a nearly optimal portfolio of (1.7). Themeaning of this result is that an investor who wants to control (1.7) for aparticular k will have the same behavior as an investor whose risk parameteris (k). See a similar result in [36] for some discrete time models.


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4 H. HATA, H. NAGAI AND S.-J. SHEU

In this paper we consider only the linear Gaussian models. But our methodmay be applied to more general (nonlinear) models. Gaussian models havepractical uses for practitioners. A simple Gaussian factor model was firstproposed in [26]. Such models have several important properties. It is mucheasier to estimate the coefficients by using linear regression (see [32]). Thishas practical applications. Tractibility is another big advantage of thesemodels. For such models, a closed-form solution for the optimal portfolioselection problem (1.1) is possible by solving a Ricatti equation, which isa matrix equation and is easier to solve [[2, 3], [1416, 25]]. For Gaussianmodels, we have some finer mathematical results [25]. In the later sections,we will see that these results will be crucial for our analysis. It is important

to consider the model when some factors cannot be observed directly. Thesolution is far away from complete. One can find some results for linearGaussian models in [18, 29, 34]. However, this is not our main concern inthis paper.

The papers [30, 31] (and also this paper) consider (1.5) [and (1.7), respec-tively] for such c (and k) that (1.5) [and (1.7)] has small probability. Thatis, we are dealing with large deviation probabilities. Therefore, we expectthat some idea from large deviation theory can be used to relate (1.1) and(1.5) [or (1.7)] that we now explain. The formal calculation given in thefollowing may be instructive to see the idea. For a given strategy , assumeZT= log X

T satisfies a large deviation principle with rate I(k, ). Formally,

this means

P log XT

T k exp(I(k, )T)

as T. The LaplaceVaradhan lemma (see [8]) implies

E[(XT)] = E[exp(log XT)] exp(T(, )),

where

(, ) = supk{k + I(k, )}.

If we want to minimize (, ) [and minimize I(k, )] over , then

inf

(, ) = inf

supk{k + I(k, )}.(1.9)

Denote

J(k) = inf{I(k, )},

() = inf{(, )}.


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If we are allowed to change the order of inf and sup on the right-hand sideof (1.9), then we obtain

() = supk{k + J(k)}.(1.10)

If J(k) is a concave function, then we expect to have the dual relation

J(k) = inf{() k}(1.11)

(see [9]). On the other hand, if we want to maximize (, ) [and maximizeI(k, )] over possible , then

sup (, ) = sup supk {k + I(k, )}.(1.12)

Denote

I(k) = sup{I(k, )},

() = sup{(, )}.

Then

() = supk{k + I(k)}.

If I(k) is concave function, then we expect to have the dual relation,I(k) = inf

{() k}.

In [30, 31], through the above intuition, the following relation was proved:

(c) = inf0


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The problem analyzed here is closely related to the problem studied in[30, 31], since

P

1

Tlog XT k

= 1P

1

Tlog XT> k

.

However, in both studies we are interested in the region of c and k suchthat probabilities in (1.5) and (1.7) are small. This means in this paper westudy the region of c such that (1.5) has large probability while in [30, 31],it is studied in the region of c that (1.5) has small probability. This explainswhy the results of [30, 31] cannot be readily applied to the calculation of thedown-side risk probabilities. We consider both of the problems for large T.

We show in this paper that the minimization of (1.7) relates to the problem(1.1) for < 0. However, the probability in (1.5) relates to problem (1.1)for 0 < < 1 as was shown in [30]. We will show that an optimal (or nearlyoptimal) portfolio for (1.7) can be derived from an optimal portfolio of (1.1)for a proper chosen < 0 that relates to k through the duality relation.This is expected, after [30, 31]. However, our result does not follow from asimple Markov inequality as in [30, 31]. We need to use some finer resultsfor (1.1), taken from [25] and some extensions given in Section 2 later. Thisis unexpected from the argument in [30, 31] and is also unexpected from thelarge deviation theory. Intuitively, when changing the order of inf and supin (1.9) to obtain (1.10), it may cause some difficulty. Mathematically, wealso want to mention another important point. The convexity of the value

of (1.1b) with respect to after taking limit for large T will be crucial inour analysis. We note that the convexity of (1.1b) does not follow for finiteT. This property is easily seen for (1.1a) from our formal derivation givenin the previous paragraph, since taking the maximum of a family of convexfunctions is a convex function.

The paper is organized as follows. In Section 2, the model will be explicitlystated and the mathematical problem of down-side risk will be formulated.The associated problems of portfolio optimization will also be described.Since the problem has different formulations in an infinite time horizon,we will careful state our problem. We also give some remarks to mentionother related problems. The main results will be stated in this section. Theseinclude the main theorem (Theorem 2.1), and several important propositions

(Propositions 2.12.7). The results are presented in a way that a proof ofTheorem 2.1 will follow using mainly results in the propositions and someminor properties. The proof of our main theorem also suggests that our mainresult will follow when we can prove the statements given in several crucialpropositions. Therefore, the approach in this paper can be applied to moregeneral models (including some nonlinear models). The problem for somenonlinear models is currently under investigation.


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In Section 3, we give the proof of our main theorem. In Section 4, weprove the propositions stated in Section 2. These propositions concern theportfolio optimization problem (1.1) and the behaviors of the solution of theBellman equation. Some results in these propositions have been obtained in[25]. They include Propositions 2.1, 2.2 and 2.3. Therefore, in the proof ofthese results, we give a sketch or modification of the arguments in [25] andmention the places in [25] where one can find the details of analysis used.Propositions 2.4, 2.5, 2.6 and 2.7 are some new results that are suggestedby the problem studied in this paper. Therefore, some more details of theproofs will be given for these propositions.

2. The problem and main results.

2.1. Down-side risk problem. We consider a market model consisting ofone bank account and m risky stocks. The interest rate of bank account aswell as the returns and volatility of stocks are affected by n economic factors.The price of bank account S0 and the price of risky stocks Si, i = 1, . . . , m ,are given by

dS0t = r(Yt)S0t dt, S

00 = s

0,(2.1)

dSit = Sit

i(Yt) dt +

n+mk=1

ik(Yt) dWkt

, Si(0) = si, i = 1, . . . , m ,(2.2)

where Wt = (Wkt )k=1,...,(n+m) is an (n + m)-dimensional standard Brown-ian motion process defined on a filtered probability space (,F, P,Ft). Thefactor process Yt with n components is described by

dYt = (Yt) dt + (Yt) dWt, Y(0) = y Rn.(2.3)

In this paper we assume that r(y), (y), (y), (y) and (y) are given by

r(y) := r,

(y) := a + Ay, (y) := ,(2.4)

(y) := b + By and (y) :=

with constants r 0, a Rm

and b Rn

. Moreover, A,B, , are m n,n n, m (n + m), n (n + m) constant matrices, respectively. In thispaper, we assume the following conditions (A):

> 0,(A1)

G := B ()1A is stable.(A2)


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Remark 2.1. The matrix G is stable if the real parts of its eigenvaluesare negative.

Consider an investor who invests at time t a proportion it of his wealthin the ith risky stock Si, i = 1, . . . , m. With t = (

1t , . . . ,

mt )

chosen, theproportion of wealth invested in the bank account is 1 t 1 where ()

denotes the transpose of a vector or a matrix. Here 1= (1, . . . , 1). We allowshort selling (it < 0 for some i) or borrowing (1

t 1< 0).

We assume the self-financing condition. Then the investors wealth, Xt ,starting with the initial capital x satisfies the equation

dXt

Xt= (1

t1)

dS0t

S0t+

m

i=1 it dSit

Sit,

X0 = x.(2.5)

This equation can be solved and we have

XT= x exp

T0

r + (a + AYt r1)

t

(2.6)

1

2t

t

dt +

T0

t dWt

.

For a finite T and given target growth rate k, we shall consider the proba-bility of minimizing the down-side risk,

infAT

P log XTT

k,(2.7)where AT is the set of all admissible investment strategies which will beprescribed in Section 2 [see (2.31)]. Here, the investor is interested in mini-mizing the probability that his wealth falls below a target growth rate. Wewill be mainly concerned with the large T asymptotics. That is,

(k) := limT

1

TinfAT

log P

log XT

T k

.(2.8)

We shall calculate (k). Connecting to this, we will obtain a (nearly) optimalstrategy for (2.7) for large T.

2.2. Portfolio optimization problem. Since it is difficult to directly calcu-late (2.7) and (2.8), in order to solve our problem we need to introduce thefollowing portfolio optimization problem of maximizing the expected utility.For < 0, we consider

supAT

E

1

(XT)

.


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This is the same as

infAT

E((XT)).(2.9)

The large T asymptotics are given by

() := limT

1

TinfAT

log E(XT), < 0.(2.10)

For (2.9), we define

J(x,y,; T) := xE(XT),(2.11)

where X0 = x, Y0 = y, and is taken from the set AT. AT will be defined

in (2.31). We then define

v(t, y) = inf ATt

log J(x,y,; T t), < 0.(2.12)

Using (2.6), we have

J(x,y,; T) = E

exp

T0

(Yt, t) dt

T

,

where (y, p) is defined by

(y, p) := r + ((y) r1)p1

2pp(2.13)

and

T= exp

T0

t dWt1

2

T0

2|t|2 dt

.(2.14)

If we assume

E(T) = 1,(2.15)

then by Girsanov theorem, we have

J(x,y,; T) = E

exp

T0

(Yt, t) dt

,(2.16)

where E

() is the expectation with respect to the probability measure P

defined by

dP

dP

FT

= T.(2.17)

We can rewrite the equation for Yt as

dYt = (Yt, t) dt + dW(t),(2.18)


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where

(y, ) = (y) + (2.19)

and

Wt = Wt

t0

s ds.(2.20)

Wt is a (n + m)-dimensional Brownian motion under P . (2.9) becomes a

stochastic control problem with criterion log J(x,y,; T) [given in (2.16)],(2.18) is the state dynamic and t is the control process. Here we note thatJ(x,y,; T) is not dependent on x. Therefore the value function is given byv(0, y) [and v(t, y) for the problem in (2.12)]. Then, by Bellmans dynamic

programming principle, v should satisfy the following Bellman equation:v

t+ infR

1

2tr(D2v) + ( + )Dv

+1

2(Dv)Dv + (, )

= 0,

v(T, y) = 0,

or, equivalently,

v

t+

1

2tr(D2v) +

+

1 ()1( r1)

Dv

+

1

2 (Dv)

I+ 1 ()1Dv+

2(1 )( r1)()1( r1) + r = 0,

v(T, y) = 0

(2.21)

(see Section VI.8 [17]). Actually, given a solution v(t, y) of (2.21), under somesuitable conditions, t defined below gives an optimal strategy of (2.9):t :=(t, Yt),(2.22)

(t, y) :=

1

1 ()(y)1((y) r1+ Dv(t, y))

(see [17, 28]). Moreover, in relation to (2.10) we shall consider the ergodictype Bellman equation which is the limiting equation of the above Bellmanequation,

= infR

1

2tr(D2) + ( + )D

+1

2(D)D + (, )

,


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or, equivalently,

=1

2tr(D2) +

+

1 ()1( r1)

D

+1

2(D)

I+

1 ()1

D(2.23)

+

2(1 )( r1)()1( r1) + r.

Remark 2.2. The portfolio optimization problem in an infinite time

horizon is closely connected with (2.10). On an infinite time horizon thecriterion to be minimized is

limT

1

Tlog E(XT)

,

A, A is a class of admissible strategies. The problem is to calculate thevalue

infA

limT

1

Tlog E(XT)

(2.24)

and obtain an optimal strategy. The Bellman equation for (2.24) is also givenby (2.23). Connecting with (2.24), we may consider the minimization of the

down-side risk probabilities in an infinite time horizon,

infA

limT

1

Tlog P

log XT

T k

.(2.25)

There are some more mathematical difficulties that arise from the problemsin an infinite time horizon. Some further discussion for this problem will begiven after we state our main theorem (Theorem 2.1).

2.3. Main results. The main idea of a stochastic control method for (2.9)or (2.10) is to solve the equations (2.21) or (2.23). From this, we can obtainthe value in (2.9) or (2.10). We may also obtain an optimal strategy for (2.9)from the solution of (2.21). We state in this subsection the relevant results.

Theorem 2.1 is our main result which also shows the connection between theproblems (2.8) and (2.9) [or (2.10)]. Some propositions are given that willbe used to prove Theorem 2.1.

Proposition 2.1 [25]. Assume (A1) and < 0. Then (2.21) has a so-lution given by

v(t, y) = 12yP(t)y + q(t)y + h(t),(2.26)


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where P(t) satisfies the Riccati differential equation:P(t) + P(t)N1P(t) + K1P(t) + P(t)K1 C

C= 0,P(T) = 0,

(2.27)

where

N1 :=

I+

1 ()1

> 0,

K1 := B +

1 ()1A,(2.28)

C :=

1 ()1A,

and q(t), h(t) satisfy the equationsq(t) + (K1 + N

1P(t))q(t) + P(t)b

+

1 (A + P(t))()1(a r1) = 0,

q(T) = 0,

(2.29)

h(t) +1

2tr(P(t)) +

1

2q(t)q(t) + bq(t) + r

+

2(1 )(a r1+ q(t))()1

(a r1+ q(t)) = 0,

h(T) = 0.

(2.30)

The proof can be found in [25]. We will give some remarks in Section 4.1.We now define the class of admissible investment strategies, AT,

AT :=

(t)t[0,T];

(2.31)

E

E

[(P(s)Ys + q(s))

+ s] dWs

T

= 1

.

This class of admissible strategies is also used in [25]. Here we use the no-tation, E(Z) := (E(Z)t)t[0,T], for the stochastic exponential of a continuous

semimartingale Z:E(Z)t := eZt1/2Zt . Therefore, T in (2.14) is equal toE(

dW)T.

Proposition 2.2 [25]. Assume (A1) and < 0. Let P(t), q(t), h(t) bedefined as in Proposition 2.1. Then the following defines an admissible strat-egy: (t) :=(t, Yt),


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(2.32) (t, y) = 11

()1[a r1+ q(t) + {A + P(t)}y]

and is optimal for the problem (2.9). Moreover,

v(0, y) = 12yP(0)y + q(0)y + h(0).

The proofs of Proposition 2.2 can be found in [25]. Some remarks will alsobe given in Section 4.1.

In the following, notation v(t, y; T), P(t; T), q(t; T), h(t; T) will be used forthe dependence of P(t), q(t), h(t) on T. Similarly, we use

v(t, y; T; ), P(t; T; ), q(t; T; ), h(t; T; ),

if we need to discuss the dependence of the functions on .We now consider (2.10). We consider a solution of (2.23) that is quadratic

in y. That is,

(y) := 12yP y + qy,(2.33)

where P is a symmetric nn matrix, and q is a vector in Rn. P and q willsatisfy the equations:

K1P + P K1 + PN1PCC= 0,(2.34)

(K1 + N1P)q + P b(2.35)

+

1 (A + P)()1(a r1) = 0.

We have the following results.

Proposition 2.3 [25]. In addition to (A1), we assume (A2). Then the following properties hold:

(i) P(t) = P(t; T) converges as T to a nonpositive definite matrixP, which is a solution of the algebraic Riccati equation (2.34). Moreover,

K1 + N1P

is stable, and P satisfies the estimate

0

esG

A()1AesG ds P 0,(2.36)

where G is given in (A2).


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(ii) q(t) = q(t; T) converges as T to a constant vector q which satis-fies (2.35). Moreover, h(t) = h(t; T) converges to a constant () definedby

() =1

2tr(P) +

1

2qq + bq + r

(2.37)

+

2(1 )(a r1+ q)()1(a r1+ q).

(iii) We obtain

limT

v(0; y)

T= limT

h(0; T)

T= ().(2.38)

The proof of Proposition 2.3 can be found in [25]. Some remarks will bealso given in Section 4.2.

We use (y), P(), q() for the dependence of (y), P , q on .

Proposition 2.4. Along with (A1) and (A2), we assume (A3);

(B, ) is controllable.(A3)

Then the pair (, ()) is the unique solution of (2.23) with = ()where (y) and () are given by (2.33) and (2.37), respectively.

The proof of Proposition 2.4 will be given in Section 4.2.

Remark 2.3. The pair (K, L), of the n n matrix K and the n lmatrix L, is said controllable if the nnl matrix (L,KL,K2L , . . . , K n1L)has rank n. We remark that the generator of Yt, Gf =

12 tr(

D2f)+ Df,is an hypoelliptic second-order operator if (A3) holds (see [20]).

Remark 2.4. It is shown in [22] that there is () such that for any (), (2.23) has a solution (y). There are only finitely many particulars where the solution (y) is quadratic in y. Under certain conditions,(2.23), for = (), has a unique solution (y) satisfying (0) = 0. Theunique pair (), () may also be characterized by the growth conditionof (y) as |y| (see [27]).

The differentiability of (y) [or (), P(), q()] will play an importantrole in the proof of our main theorem (Theorem 2.1). We have the followingresults.

Proposition 2.5. Under assumptions (A1) and (A2), the following re-sults hold:


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(i) () and P(), q() are twice differentiable with respect to .(ii) () is convex with respect to .

The proof of Proposition 2.5(i) [and (ii)] will be given in Section 4.3 (andSection 4.4).

Similar to (2.32), we use () to define

(y) =1

1 ()1((y) r1+ D(y))

(2.39)

=1

1 ()1[a r1+ q() + {A + P()}y].

Now we also define(y) := (y) +

(y) + D(y)

= b + By +

1 ()1(a + Ay r1)

(2.40)+ N1(P()y + q())

= (K1 + N1P())y + f,

where

f := b +

1 ()1(a r1) + N1q.

Define

u = u(y) := {D(y) + (y)}

=

(P y + q)(2.41)

+

1 ()1((P y + q) + a + Ay r1)

.

We will show

E

E

u dW

t

= 1, t > 0.(2.42)

Then we define a new probability measure P bydPdP

Ft

:= E

u dW

t

.(2.43)

Define Wt byWt = Wt t

0u(Ys)

ds.(2.44)


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Then Wt is a Brownian motion under the probability measure P, and Ysatisfies

dYt = (Yt) dt + dWt.(2.45)Remark 2.5. We compare (y) in (2.40) and (, ) in (2.19),

(y) = (y, (y))+ (P()y + q()).

As shown in (2.16), (y, (y)) is used to change the measure to derive theuseful expression ofJ(x,y,; T). Because of the integral from 0 to T in (2.16)

when t = (Yt), the expectation may grow exponentially and will causedifficulty in the analysis. The difference of (y) and (y, (y)) is made totake care of this. This gives another measure change, hence another term isadded to (y, (y)) which leads to (y). A similar idea is used in [23]. Inthe proof of our main theorem (Theorem 2.1), we will also see some uses ofP.

Under P, Yt is a Gaussian process. The variance of Yt is given by U(t),U(t) =

t0

e(ts)(K1+N1P)e(ts)(K1+N

1P) ds,

and its mean m(t) is the solution of the following equation:

m(t) = (K1 + N1P)m(t) + f.

We show in the next proposition that K1 + N1P is stable under as-

sumption (A2); then Yt is ergodic under P.Proposition 2.6. Under assumptions (A1) and (A2), K1 +

N1Pis stable. There are c1() > 0, c2() > 0 and a positive definite n n matrixK such that

yK(y)c1()|y|2 + c2(),(2.46)

where (y) is given by (2.40).

The proof is given in Section 4.5.

Proposition 2.7. Under assumptions (A1) and (A2) we obtain

lim

() = r.(2.47)


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For the proof, see Section 4.6. This result gives the range of k in (2.8)that our main result holds (Theorem 2.1).

We can now state our main theorem. Its proof, given in next section, isbased on the results in Propositions 2.12.7.

Theorem 2.1. Assume (A1) and (A2). Then, for all r < k < (0),we have the following:

(k) = inf


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Hence, to avoid confusion, we may use tT,k for [k]t . Therefore, for = (k),E((X

T,k

T )) = infAT

E((XT)).

tT,k is an optimal strategy for (2.9) [= (k)] but may not be an optimalstrategy for (2.7). This is the reason we say tT,k is nearly optimal for (2.7)when T is large, since the value using this strategy is close to the optimalvalue when T is large.

On the other hand, we can use () in (2.39) to define

(t) = (Yt).

We may expect that this will also give a nearly optimal strategy. That is, ina sense,

E((XT ))(2.52)

is close to

infAT

E((XT)),

ifT is large. There are two problems when one wants to prove this rigorously.For the first problem, it is not easy to show() is an element ofAT. For thesecond problem, it may happen that (2.52) becomes infinite for some finiteT. See [14] for a study of a model where there is one stock in the market.

When this happens, the problem (2.24) may not have a solution. That is,there is no optimal strategy for (2.24). However, it is shown in [14] thatsome modification of () gives a nearly optimal strategy. Such behavioralso indicates that (2.25) may be more difficult to treat than (2.8). However,if we assume (A1)(A3) and the following condition,

P()()1P() < A()1A,

then it is proved in [25] (Theorem 2.3) that

limT

1

Tlog E(X

T ) = ().

Using this and following the same argument as in Theorem 2.1, we can obtain

the upper estimate for down-side risk probability. For the lower estimate,the same argument in Theorem 2.1 can be applied. In conclusion, we havethe following result; its proof is omitted.

Theorem 2.2. Assume (A1)(A3). Let r < k < (0) and (k) < 0 bethe unique number satisfying ((k)) = k. Assume

P((k))()1P((k)) < A()1A.


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Define k(t) := (k)(Yt).Then k() AT for any T and k on a sufficiently large time horizon T isa nearly optimal strategy for the problem (2.7). Namely,

(k) = limT

1

Tlog P

log XkT

T k

.

In (2.25) we define A to be a family consisting of processes t such that|[0,T], the restriction of t to [0, T], is inAT for all T. Then

k() A and

is an optimal strategy for (2.25).

3. Proof of Theorem 2.1. In this section,we shall give a proof of our maintheorem using results in Propositions 2.12.7. The proof of the propositionswill be given in Section 4.

From Proposition 2.5(ii), is convex. Let us consider(k) := inf r.

Since is smooth, we see that (k) is strictly concave, nondecreasing andsatisfies

(k) =

0, if k (0),((k)) (k)((k)), if r < k < (0),

where (d) < 0 is such that

((d)) = d (r,

(0)). In Section 4.4, we willshow () > 0 for < 0. Therefore, (d) is uniquely defined. Moreover, is continuous on (r,(0)).

We now take a small > 0. Then k > r. Suppose < 0 attains thefollowing: (k ) := inf


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or, equivalently,

() = 12 tr(D2(y)) + (y)

D(y) + V0,(3.2)

where (y) is defined in (2.40) and

V0 = V0(y)

:= 1

2D(y)N1D(y)(3.3)

+

2(1 )(a + Ay r1)()1(a + Ay r1) + r.

By differentiation of (3.2) with respect to , we have

() =1

2tr(D2(y)) + (y)

D(y)

+1

(1 )2()1(a + Ay r1)D

+1

2(1 )2(D)(y)()1D(y)

+1

2(1 )2(a + Ay r1)()1(a + Ay r1) + r,

where =

. That is,

() =1

2tr(D2(y)) + (y)

D(y) + V1,(3.4)

V1 = V1(y) :=1

2(1 )2{D(y) + (a + Ay r1)}()1

(3.5){D(y) + (a + Ay r1)}+ r.

Furthermore, from (3.2) and (3.4), we can obtain

() ()(3.6)

= 1

2tr(D2( )(y)) +

(y)D( )(y) + V

2,

where V2 is defined by

V2 = V2(y) = 12 |u|

2;(3.7)

u() is in (2.41). Indeed, from (3.2) and (3.4),

() () = 12 tr(D2( )(y)) + (y)

D( )(y) + V0 V1.


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By (3.3) and (3.5), a straightforward calculation shows that

V0 V1 = 1

2D(y)D(y)

1 {D(y) + (a + Ay r1)}()1D(y)

1

2

1

2{D(y) + (a + Ay r1)}

()1{D(y) + (a + Ay r1)}

= 1

2

|u|2,

where the last equality follows from (2.41). Hence we have (3.7).For AT, we have by (2.6),

XT= x exp

T0

r + (a + AYt r1)

t 1

2t

t

dt +

T0

t dWt

.

Then

log XTT

=log x

T+

1

T

T0

t dWt

+1

T T

0

1

2t

t+ {t (a + AYt r1) + r}

dt.

Recalling Wt defined by (2.44), we can rewrite the last relation aslog XT

T=

log x

T+

1

T

T0

t dWt

1

2T

T0

t

1

1 ()1(D(Yt) + a + AYt r1)

t

1

1 ()1(D(Yt) + a + AYt r1)

dt

+1

T T

0 1

2(1 )2

(D(Yt) + a + AYt r1)

()1(D(Yt) + a + AYt r1) + r

dt.

From (3.5) and with some calculation, we can rewrite this as

log XTT

=log x

T+

1

T

T0

1

1 {()1(D(Yt)


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+ a + AYt r1)} dWt+

1

TlogE

1

1 ()1(D(Y)(3.8)

+ a + AY r1)

dW

T

+1

T

T0

V1(Yt) dt.

Define the following events:

A := ; log XT

T k,

A1,T :=

;

1

T

T0

V2(Yt) dt () ()

,

A2,T :=

;

1

T

T0

u(Yt) dWt

and

A3,T :=

;

log x

T+

1

T

T0

V1(Yt) dt () +

2

,

A4,T := ; 1T T

0

1

1 {()1(D(Yt)

+ a + AYt r1)}

dWt 4

,

A5,T :=

;

1

TlogE

1

1 ()1

(D(Y) + a + AY r1)

dW

T

4

.

From (3.1) and (3.8), we see that

A3,T A4,T A5,TA.(3.9)

Recall that P is the probability defined by (2.43). By using (3.6), (2.45)and Itos formula, we have

( )(YT) ( )(y) = (() ())T

T0

V2(Yt) dt

+

T0

D( )(Yt) dWt.


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We apply Chebyshevs inequality:

P(Ac1,T) 12E() () 1T

T0

V2(Yt) dt

2

1

2T2E( )(YT) ( )(y)

T0

D( )(Yt) dWt2.Noting that (y), and (y) are quadratic in y, then

|( )(y)| K(1 + |y|

2

),|D( )(y)| K(1 + |y|).

Therefore, we have

E( )(YT) ( )(y) T0

D( )(Yt)(Yt) dWt2K

1 + |y|4 + T + E[|YT|4] + T

0

E[|Yt|2] dt d1(){|y|

4 + 1 + (|y|2 + 1)T}

for some d1

() > 0 where the last inequality follows from Lemma 3.1 below.Therefore, we see that

P(Ac1,T) d1(){|y|4 + 1 + (|y|2 + 1)T}2T2and P(Ac1,T) ,(3.10)provided T is sufficiently large.

By using Chebyshevs inequality again, we have

P(Ac2,T) 1

2

T2E

T

0

u(Yt) dWt

2

=1

2

T2ET

0

|u(Yt)|2 dt.

By (2.41) and using |D(y)| K(1 + |y|), we obtain

P(Ac2,T) K2T2

T +

T0

E[|Yt|2] dt

d2()(|y|2 + 1)

2T,


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for some d2() > 0 where the last inequality follows from Lemma 3.1 below.Therefore, we see that P(Ac2,T) ,(3.11)provided T is sufficiently large.

Using a similar argument, we can showP(Ac4,T) .(3.12)We now consider P(Ac3,T). By (3.4) and Itos formula,

d(Yt) = V1(Yt) dt + () dt + (D(Yt))

d

Wt.

From this, we haveT0

V1(Yt) dt = (Y0) (YT) + ()T +

T0

(D(Yt)) dWt.

Then, A3,T is the same as;

log x

T+

(Y0) (YT)

T+

1

T

T0

(D(Yt)) dWt

2

.

Now we can use the same argument as above to get

P(Ac3,T) ,(3.13)

provided T is sufficiently large.For A5,T, we haveP(Ac5,T) e/4T

EE 11

()1(D(Y)

+ a + AY r1)

dW

T

e/4T.

Then we have P(Ac5,T) ,(3.14)if T is sufficiently large.

Hence, from (3.7), (3.9), (3.10), (3.11), (3.12), (3.13) and (3.14), we have

P(A) = EexpT0

u(Yt) dWt 1

2

T0|u(Yt)|

2 dt

; A


25/40


= EexpT0

u(Yt) dWt+ T

0V2(Yt) dt

; A exp[(() () 2)T] P(A1,T A2,T A) exp[(() () 2)T] P(A1,T A2,T A3,T A4,T A5,T) exp[(() () 2)T] (1 5).

The estimate of P(A) is uniform in . Therefore, we see that

(k) limT

1

Tlog{exp[(() () 2)T] (1 5)}

= (k ) 2.By continuity of on ((), (0)) and sending to 0, we obtain

(k) inf


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Since () is bounded, () r as (Proposition 2.7), we see

() k , .

In particular,

inf


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By (4.13), we have

(0) =1

2tr

dP

d(0)

+

dq

d(0)b +

1

2(a r1)()1(a r1) + r.

Using the above expression for dPd(0), we can show

BdP

d(0) +

dP

d(0)B = A()1A.

Multiplying this by (B)1 on the left and B1 on the right, we have

(B)1dP

d

(0) +dP

d

(0)B1 = (B)1A()1AB1.

Then

(B)1dP

d(0)b b

= 1

2(AB1b)()1AB1b,

dq

d(0)b +

1

2(a r1)()1(a r1)

=1

2(AB1b (a r1))()1(AB1b (a r1)).

An expression of (0) as in (2.51) follows.Finally, we show

inf


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hence

inf 0. Moreover, if we take as 0 < < 2pc1(), then we have

d|Yt|2pet 2pet|Yt|

2(p1)Yt dWt+ et|Yt|

2(p1)[p{2c1()|Yt|2 + 2c2() + c3}+ |Yt|

2] dt

2pet|Yt|2(p1)Yt d

Wt + etM dt,where

M := supy|y|2(p1)[p{2c1()|y|

2 + 2c2() + c3}+ |y|2].

Therefore, we obtain

|Yt|2p |y|2p +

M

(1 et) + 2p

t

0e(ts)|Ys|

2(p1)Y

s dWs.

From this, by an argument using a stopping time, we obtain (3.16) withM = M /.

Remark 3.1. The proof of Theorem 2.1 is based on the results in Propo-sitions 2.12.7. It is interesting to verify these results for more general non-Gaussian models. We make the following observation. If we assume:


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(i) ,,, are globally Lipschitz;(ii) 1||

2 (y) 2||2, 1, 2 > 0;

(iii) 1||2 (y) 2||

2, 1, 2 > 0; then there is () such that

(2.23) has a solution for () (see Theorem 2.6 in [22]). Moreover, undercertain conditions, there is a unique solution (y) with (0) = 0 for =() (see Theorem 3.8 in [22]). Assume is positive, and for Gaussiancases studied here that satisfy the conditions (A1), (A2), (A3), () =(), (y) = (y), which are defined in Proposition 2.3. It is an interestingproblem to find conditions on coefficients that prove other propositions.

4. Proof of propositions.

4.1. Finite time horizon problem. In this subsection, we prove Propo-sitions 2.1 and 2.2. We follow closely the arguments of Kuroda and Nagai[25].

First of all, we attack the finite time horizon problem (2.9). Then thesolution v of the Bellman equation (2.21) can be expressed as quadraticfunction such that

v(t, y) = 12yP(t)y + q(t)y + h(t),

provided that equation (2.27) has a solution. Here q and h are solutions of(2.29) and (2.30), respectively. We recall the following result (5.2) in [13],Theorem IV. If < 0, then we see that (2.27) has the unique solution P(t)

0. See also Remark 1 of Section 2 in [25], but notice that and the solutionP(t) of (2.27) correspond to 2 and

2P(t) in [25], respectively. Therefore,

as in the proof of Theorem 2.1 in [25], we obtain Propositions 2.1 and 2.2.

4.2. Asymptotics as T. In this section we prove Propositions 2.3and 2.4. Here we recall the following theorem (see Theorem 4.1 and Lemma5.2 in [38]).

Theorem 4.1 [38]. Assume that N > 0 and (K1, ) is stabilizable, thenfor the solution Q(t) = Q(t; T) of

Q(t) + K1Q(t) + Q(t)K1 Q(t)N1Q(t) + CC= 0,Q(T) = 0.(4.1) limTQ(t; T) = Q, and Q satisfies the algebraic Riccati equation,

K1Q + QK1QN1Q + CC= 0.(4.2)

Moreover, if (C, K1) is detectable, then K1 N

1Q is stable and thenonnegative definite solution of (4.2) is unique.


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Remark 4.1. The pair (L, M) of the n n matrix L and the n lmatrix M is stabilizable if there exists a ln matrix K such that LMKis stable. The pair (L, F) of the l n matrix L and the n n matrix F iscalled detectable if (F, L) is stabilizable.

Proof of Proposition 2.3(i) and (ii). Let us set Q(t) = P(t).Then we see that Q satisfies (4.1). Let K1, C be given in (2.28). TakeK= ()1A. Then K1K= B

()1A = G is stable. Note

also that if we set K2 :=

1(1)

, then K1 CK2 = G is stable [G

is given in (A2)]. Therefore, we see that (K1, ) is stabilizable, (C, K1) isdetectable and Theorem 4.1 applies. Then we can follow the arguments of

Proposition 2.2 in Kuroda and Nagai [25]; we omit the details here.

Proof of Proposition 2.3(iii). Note that P(t; T) is uniformly boundedwith respect to t and T (see Remark 1 of Section 4 in [25]). Moreover,since K1 + N

1P(t; T) converges to a stable matrix K1 + N1P

as T, we can show by (2.29) that q(t; T) is uniformly bounded withrespect to t and T. See Lemma 4.4 in [25]. Therefore we obtain (2.38).

Proof of Proposition 2.4. Let us assume (A1)(A3). Then we havethe solution P of (2.34) and q of (2.35). Moreover, the pair ((y), ()) of(y), defined by (2.33) and () defined by (2.37), satisfies (2.23) (cf. Section2 in Kuroda and Nagai [25]). Here we note that from (A2), K1 + N

1P

is stable (by Theorem 4.1). Moreover, from (A3) we see that the variance ofYt under P [defined in (2.43)] is nondegenerate (see Lemma 5.1 of Kurodaand Nagai [25]). Therefore, we see that Yt given by (2.45) is P-ergodic. Therest of the proof follows closely the arguments of Theorem 3.8 in Kaise andSheu [22] and is omitted here.

4.3. Differentiability with respect to HARA parameter . In this subsec-tion, we shall prove Proposition 2.5(i). Let us first note that the Riccatidifferential equation (2.1) can be solved by considering a Hamiltonian sys-tem. Indeed, introduce a Hamiltonian matrix H defined by

H =K1 N1

C

C K1 ,(4.3)

and consider the ordinary differential equationU(t)V(t)

= H

U(t)V(t)

, 0 t T,

U(0)V(0)

=

En0

.(4.4)

See Chapter V in [1]. Note that U and V are nn matrix valued functionson 0 t T, and En is the nn unit matrix. Then it is known that U(t) is


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invertible, and W(t) := V(t)U(t)1 is the solution to the Riccati differentialequation

W(t) = K1W(t) + W(t)K1W(t)N1W(t) + CC,

W(0) = 0.(4.5)

Then we see that, by setting P(t) := P(T t; T), we have P(t) = W(t).Lemma 4.1. The solutionP(t) := P(t; T; ) to the Riccati equation (2.27)

is inC1-class with respect to .

Proof. The Hamiltonian matrix H defined by (4.3) is smooth with

respect to and so is the solution (U(t), V(t))

of (4.4). Moreover, U(t) isinvertible. Therefore, U(t)1 is in C1-class and

U(t)1

=U(t)1

U(t)

U(t)1.

Thus we see that W(t) = V(t)U(t)1 is in C1-class with respect to . Hencewe conclude our present lemma.

Now, let us rewrite (2.27) as

P(t) + (K1 + N1P(t))P(t) + P(t)(K1 + N

1P(t))(4.6)

P(t)N1P(t)CC= 0.

Then by differentiating (4.6) with respect to , we obtain

d

dt

P

+ (K1 + N

1P(t))P

+P

(K1 + N

1P(t))(4.7)

+1

(1 )2(P(t) + A)()1(P(t) + A) = 0.

Then we obtain the following lemma.

Lemma 4.2. Assume (A1) and (A2). Then the solutionP

(t; T; ) of(4.7) converges to dPd which satisfies

(K1 + N1P)

dP

d+

dP

d(K1 + N

1P)

(4.8)

+1

(1 )2(P + A)()1(P + A) = 0.


32/40


33/40


(4.10) ()1(P + A)es(K1+N

1P) ds.

On the other hand, we have

P(+ ) P() = limT

{P(t; T; + ) P(t; T; )}

= limT

+

P

(t; T; u) du

=

+

P

(u) du.

From (4.10), (P) is continuous with respect to . Therefore, we see that P

is differentiable with respect to , and dPd() = (P )().

As for differentiability of q with respect to , we can see this is similarto Lemma 4.2. Indeed, (2.29) is a linear equation and its coefficients areall in C1-class with respect to . Therefore, the solution q(t) of (2.29) is inC

1-class with respect to , and we have

d

dt

q

+ (K1 + N

1P(t))q

+

(K1 + N1P(t))

q(t)

+P

b +

1

(1 )2(P(t) + A)()1(a r1)(4.11)

+

1

P

()1(a r1) = 0.

Thus we have the following lemma, similar to Lemma 4.2.

Lemma 4.3. Under assumptions (A1) and (A2), as T, q(t; T; ),

the solution of (4.11) converges to dqd which satisfies

(K1 + N1P)

dq

d+

,

dP

d, P , q

= 0,(4.12)

where

, dPd

, P , q := 1(1 )2

()1(A + P) + N1 dPdq

+dP

db +

1

(1 )2(A + P)()1(a r1)

+

1

dP

d()1(a r1).


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Differentiability of () is directly seen from (2.37). Indeed, we have

d

d=

1

2tr

dP

d

+ q

dq

d+ b

dq

d+ r

+1

2(1 )2(a r1+ q)()1(a r1+ q)(4.13)

+

1 (a r1+ q)()1

dq

d.

The following lemma is a direct consequence of (4.9), (4.12) and (4.13).

Lemma 4.4. Under assumptions (A1) and (A2),dP

d,

dq

d and

d

d are dif-ferentiable with respect to .

Proof. Differentiability of dPd is seen by looking at (4.9). As fordqd,

from (4.12) we obtain

dq

d= [(K1 + N

1P)]1

,

dP

d, P , q

,

and so it turns out to be differentiable. From these facts and ( 4.13), differ-

entiability of dd follows.

4.4. Convexity of. In this subsection, we shall show Proposition 2.5(ii).

Proof of Proposition 2.5(ii). Note that K1 + N1P is stable

under assumption (A2). In the previous subsection, namely the proof ofProposition 2.5(i), we have shown in Lemma 4.4 that under assumptions(A1) and (A2), P , q and are twice differentiable with respect to and sois . Recall that

d

d() =

1

2tr(D2(y)) + (y)

D(y) + V1,

where := , (y) is given by (2.40) and V1 is defined by (3.5). Moreover,

setting := 2

2 , we have

d2

d2() =

1

2tr(D2(y)) + (y)

D(y) +

(y)

D(y)

+1

(1 )3{D(y) + (a + Ay r1)}

()1{D(y) + (a + Ay r1)}


35/40


+ 1(1 )2

(D)(y)()1

{D(y) + (a + Ay r1)}.

Using

(y) =1

(1 )2()1{D(y) + (a + Ay r1)}

+ N1D(y),

we obtain

d2

d2

() =1

2

tr(D2(y)) + (y)D+ (y; ),

where

(y; ) := (D)(y)(I()1)D(y)

+1

(1 )3{(1 )D(y) + D(y) + (a + Ay r1)}

()1{(1 )D(y) + D(y) + (a + Ay r1)}.

Note that |(y; )| K(1 + |y|2). From (3.16) we can see that

d2

d2() = limT

1

T

E

T0

(Ys; ) ds

0.

Then K satisfies the following equation:

G

K+ KG =I.

Therefore, we have

KGy,y+ y,KGy= y, y,Gy,Ky= 12y, y.

Since

(y) = Gy + f

[see (2.40)], we can deduce (2.46) after some calculation. Here K= K, c1() =1/4, and c2() = |Kf|

2.


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4.6. Asymptotics as . In this subsection we shall consider asymp-totic behavior of dd() as , and obtain Proposition 2.7.

Proof of Proposition 2.7. We first consider P() which is a solution

of the algebraic Riccati equation (2.34). We then note that dPd 0 holds, and

P is bounded because of (4.9) and (2.36). We set P() := limP().Now we rewrite (2.34) as

(K1 K)P + P(K1 K) + (

P + NK)N1(P + NK)(4.14)

KNKCC= 0,

where

K:=1

1 ()1A.(4.15)

Noting that

N = I ()1,

lim

K= 0, lim

K1 = G,

lim

N1 = I()1 := N(),lim

(KN K+ CC) = A()1A,

where G = B

(

)1

A [see (A2)]. We obtainGP() + P()G

(4.16)+ P()N()P()A()1A = 0.

Moreover, we rewrite (4.16) as

(G + N()P())P() + P()(G + N()P()) (P()N()P() + A()1A) = 0.

We consider

d

dt

et(G+N()

P())

P()et(G+ N()

P())

and using the above relation, we can show0

es(G+N()

P())

P()

N()P()es(G+ N()P()) ds(4.17)P().


37/40


Here we use P() 0. Since G is stable,

(G + (N()P()), (N()P()))is stabilizable which means that (N()P(), G + N()P()) isdetectable. Therefore, noting that (N())2 = N() and that

0es(G+

N()P())

P()N()P()es(G+ N()P()) dsis bounded because of (4.17), we see that G +

N()

P() is stable(see [39], Proposition 3.2). Now noting that

K1 K= G, dd

(K1 K) = 0,

NK= ()1A,d

d(NK) = 0,

KN K+ CC= A()1A andd

d(KNK+ CC) = 0.

Then, by differentiating (4.14) with respect to , we obtain

(K1 + N1P)

dP

d+

dP

d(K1 + N

1P)

(4.18)

+ 1(1 )2

(P + N K)()1(P + N K) = 0.

Set (dPd)() =limdPd . Then, sending to in (4.18), we see that

(G + N()P())dPd

()

+

dP

d

()

(G + N()P()) = 0.Since G +

N()

P() is stable, we see that

dPd

()

= 0.(4.19)

Set q() =lim q(). As for q(), sending to in (2.35), we have

(G + N()P())q() + bP() (A + P()

)()1(a r1) = 0,


38/40


and so

q() = {(G + N()P())}1 [(A + P()

)()1(a r1) bP()].

Moreover, setting ( dqd)() = limdqd, we see by (4.12) that

(G + N()P()) dqd

()

= 0.

Since G +

N()

P() is stable, we have

dqd() = 0.(4.20)The present proposition is directly seen by using (4.13), (4.19) and (4.20).

Acknowledgments. The authors would like to thank the referees for help-ful comments and suggestions. The first author is a postdoctoral fellow atInstitute of Mathematics, Academia Sinica, Taiwan. Therefore, he is thank-ful to Academia Sinica for constant support.

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H. HataS.-J. Sheu

Academia Sinica

Nankang, Taipei 11529

Taiwan

E-mail: [email protected]

[email protected]

H. NagaiOsaka University

Toyonaka

560-8531, Osaka

Japan

E-mail: [email protected]
http://www.ams.org/mathscinet-getitem?mr=1702984http://www.ams.org/mathscinet-getitem?mr=2002529http://www.ams.org/mathscinet-getitem?mr=1093001http://www.ams.org/mathscinet-getitem?mr=0239161http://www.ams.org/mathscinet-getitem?mr=0770574mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.ams.org/mathscinet-getitem?mr=0770574http://www.ams.org/mathscinet-getitem?mr=0239161http://www.ams.org/mathscinet-getitem?mr=1093001http://www.ams.org/mathscinet-getitem?mr=2002529http://www.ams.org/mathscinet-getitem?mr=1702984

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