Khai triển trực giao của các hàm ngẫu nhiên

8/2/2019 Khai trin trc giao ca cc hm ngu nhin

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ti Khai trin trc giao ca hm ngu nhin

2Lun vn thc s tonhc Trn ThVn Anh

LI MU

Xc Sut Thng K l lnh vc Ton hc ng dng, n i hi mt cs

ton hc su sc. Ngy nay cc m hnh Xc Sut thc sc ng dng rng

ri trong Khoa Hc T Nhin cng nh Khoa Hc X Hi.

Trong lun vn ny, nghin cu v khai trin trc giao ca hm ngunhin. V mt l thuyt chng c nhiu tnh cht th v lin h vi cc qu trnh

ngu nhin khc. V mt ng dng chng trthnh cng c ton hc c hiu lc

cho nhiu vn trong cc lnh vc khc nhau nh ton hc, vt l, sinh hc, c

hc, khoa hc tri t, kinh t

Lun vn ny gm 3 chng :

Chng 1 : MT S KIN THC CBN

Trong chng ny nghin cu v nhc li kin thc cbn cn cho lun

vn ny, cn c k cc khi nim v nm vng cc kt qu nhc mu

bng vic gii thiu khng gian Hilbert gm cc bin ngu nhin bnh phng

kh tch vi v hng l hi p phng sai ca hai bin ngu nhin, dng php

chiu trc giao xy dng php xp x tuyn tnh v lp phng trnh don,

tip theo nu khi nim k vng c iu kin v chng t rng k vng c iu

kin l don tt nht. Khai trin chnh tc ca qu trnh ngu nhin cng c

nghin cu trong chng ny. Ngoi ra cn nghin cu qu trnh Wiener v tch

phn Ito l hai khi nim quan trng khi nghin cu v qu trnh ngu nhin. y

l nhng khi nim cbn v l cs nghin cu nhng vn tip theo.

Chng 2 : A THC HERMITE V KHAI TRIN FOURIER

HERMITE


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Chng ny nghin cu cc nh ngha, cc tnh cht v b ca a

thc Hermite v tnh cht ca khai trin Fourier Hermite. Mt vi bng

dng c chng minh trong chng ny l cng c chnh ta s dng tip cho

chng sau.

Chng 3 : QU TRNH NGU NHIN DNG HERMITE

Chng ny mrng a thc Hermite ca chng 2 l nghin cu qu

trnh ngu nhin dng Hermite. Bt u khi nim v qu trnh ngu nhin dngHermite. Sau mrng khi nim l xc nh hm Hermite chun suy rng, s

dng chng thu c t p trc chun y trong ( )2L R v ( )2 nL R . Cui

cng nghin cu v nu c mt sc tnh ca vi phn ngu nhin i vi qu

trnh ngu nhin dng Hermite.


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MC LC

Trang

Li cm n . 1

Li ni u . 2

Mc lc ... 4

CHNG I MT S KIN THC CBN. 7

1.1 Khng gian 2( , , )L F P .. 7

1.1.1 Bin ngu nhin 7

1.1.2 nh ngha 7

1.1.3 nh ngha .... 8

1.1.4 Tnh cht 9

1.1.5 nh l (nh l v php chiu trong khng gian Hilbert) 9

1.1.6 Tnh cht ca php chiu ... 12

1.1.7 Php xp x tuyn tnh trong L2 12

1.1.8 Phng trnh don . 13

1.1.9 K vng c iu kin v don tt nht trong L2 14

1.2 Khai trin chnh tc ca qu trnh ngu nhin .16

1.2.1 Qu trnh ngu nhin biu din di dng tng cc hm

ngu nhin cbn 16

1.2.2 Khai trin chnh tc qu trnh ngu nhin 18

1.2.3 a qu trnh ngu nhin v dng chnh tc 20

1.2.4 Mt s khai trin chnh tc c bit 22

1.3 Cstrc giao v trc chun trong khng gian Hilbert 25


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1.3.1 nh ngha (Trc giao v trc chun) 25

1.3.2 nh ngha ( Cs) 25

1.3.3 nh ngha ( Cstrc giao v trc chun ) 26

1.3.4 nh ngha ( Php chiu trc giao ) 26

1.4 Qu trnh Wiener 27

1.4.1 nh ngha ( Qu trnh Wiener ) 27

1.4.2 Cc tnh cht qu trnh Wiener v o 27

1.4.3 Qu trnh Wiener n - chiu 37

1.5 Tch phn Ito 391.5.1 nh ngha .. 39

1.5.2 Cc tnh cht cbn ca tch phn Ito 40

1.5.3 Tch phn Ito nhiu chiu 43

1.5.4 Vi phn ngu nhin ca hm hp, cng thc Ito ......... 44

CHNG 2 A THC HERMITE V KHAI TRIN FOURIER

HERMITE

2.1a thc Hermite ..48

2.1.1 nh ngha ..482.1.2 Lin h gia a thc trc giao v a thc Hermite 492.1.3 o hm ca a thc Hermite 502.1.4 Cc b ca a thc Hermite 53

2.2 Khai trin Fourier Hermite ca hm bin ngu nhin Gauss 572.2.1 Khai trin Fourier Hermite 57

2.2.2 Tnh cht 58

CHNG 3 QU TRNH NGU NHIN DNG HERMITE 60

3.1 Khi nim vqu trnh ngu nhin dng Hermite 60

3.1.1 nh ngha ..60


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3.1.2 Cc v d 60 3.2 Tp trc chun y trong ( )

2

L R v ( )2 n

L R ... 62

3.2.1 nh ngha 623.2.2 Cc tnh cht 623.2.3 nh ngha . 643.2.4 Tnh cht 65

3.3 Mt sc tnh ca vi phn ngu nhin 66

3.3.1 nh ngha ..663.3.2 nh l 673.3.3 B 673.3.4 H qu 693.3.5 Cc tnh cht ca qu trnh dng Hermite 70KT LUN 74

TI LIU THAM KHO.. 75


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CHNG 1

MT S KIN THC CBN

1.1 KHNG GIAN 2( , , ) L F P

Phn ny gii thiu khng gian cc bin ngu nhin bnh phng kh tch

L2( , ,F P )

1.1.1 BIN NGU NHIN

Bin ngu nhin l i lng m gi tr ca n ph thuc vo kt qu ca

th nghim . Ta nh ngha chnh xc bin ngu nhin l :

Xt php th ngu nhin vi tp v - i s F cc bin cBin ngu nhin l nh x ( ): ,X R = + sao cho:

( )( ) ( ){ }\ F, X x X x x R =

hoc :

( ) ( ){ }1 \ , X B X B F = B B

vi B l tp cc tp Borel trong R .

Ta ch xt nhng tp B sao cho ( )1X B l bin c, tc F, khi lp tt

c cc bin c ( )1X B l lp bin c cm sinh bi bin s ngu nhin ( )X .

1.1.2 NH NGHA

Ta xt khng gian xc sut ( ), ,F P v l p cc bin ngu nhin bnh

phng kh tch c nh ngha trn v tha mn iu kin :


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2 2 ( ) ( ) EX X P d

= <

Khi , ta c :

( )2 2 2 , , E cX c EX c R X =

Mt khc:

( )2 2 22 2 X Y X Y + + = = . (1.1)

Khng gian 2 ( , , ) L F P l t p cc lp tng ng vi tch v hng

c nh ngha theo cng thc (1.1), mt khc v mi lp tng ng c

xc nh duy nht bng cch ly mt phn t bt k no ca lp lm i din

nn ta vn dng k hiu X, Y ch cc phn t ca ( )2 , , L F P , ta c th dng

ngn gn 2L v vn gi l nhng bin ngu nhin bnh phng kh tch v ta

ch rng nu ch c X th hiu rng X l i din cho c mt lp cc bin ngu

nhin tng ng vi X.

1.1.3 NH NGHA

S hi t trong L2 l s hi t bnh phng trung bnh vit l2L

nX X

ngha l, dy cc phn t { }nX , { }2

nX L c gi l hi tn X nu v ch

nu :


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22 : 0n n X X E X X = khi n

xy dng tnh y ca L2 l khng gian Hilbert ta cn phi xy dng

tnh y ca L2 ngha l nu 2 0m nX X khi ,m n th tn ti2X L sao

cho:2L

nX X

Ta xt tnh cht :

1.1.4 TNH CHT

Nu 2nX L v 1 2 nn nX X + ; n = 1, 2, 3 th tn ti mt bin

ngu nhin X trn ( , , )F P sao cho2L

nX X .

Chng minh:

Chn 0X = 0

t Xn : = 11

j j

j

X X

=

, khi theo bt ng thc Cauchy Schward, ta c :

, . X Y X Y < >

E ( 11

j j

j

X X

=

) = 11

j j

j

E X X

=

11 1

2 jj jj j

X X

= =

<

T, suy ra tn ti 11

limn

j jn

j

X X

=

v gii hn hu hn.

Nh th

11

lim ( ) limn

j j nn n

j

X X X

= = tn ti.

1.1.5 NH L (nh l v php chiu trong khng gian Hilbert)

Nu A l mt khng gian con ng ca khng gian Hilbert H v H

th:

a) Tn ti duy nht mt phn t 'x A sao cho ' inf y A

x x y

=


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b) ' A v ' inf y A

x x x y

= nu v ch nu ' A v ( )' x x A

x c gi l chiu (trc giao) ca x ln A, vit l ' : A x P x=

nh l ny c gi l nh l v php chiu trc giao.

Chng minh:

a) Nu 2y A

: inf d x y

= th tn ti mt dy { }ny , ny A sao cho

2 0n

y x .

Hn na, vi k, l bt k thuc khng gian Hilbert, theo quy tc ng cho hnh

bnh hnh ta c :

22 2 22k l k l k l + + = +

Do , xt ,m n

y x A y x A

Ta c:

2 2 2 2

2m n m n m ny x y x y x x y y x y x + + + = +

tc l:

2 2 2 22 2

m n m n m n y y x y y y x y x + + = +

Mt khc, v:

( ),

2m n

y yA

( )2

2 220 4 22

m n

m n m ny y y y x y x y x = + +

( )2 24 2 0m nd y x y x + + khi ,m n


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T theo tiu chun Cauchy, ' H sao cho ' 0n

y x v v A ng nn

'x A v v tnh lin tc ca tch v hng nn :

22

' limn

n x x x y d

= =

chng minh tnh duy nht ca x ta gi s c 'y A sao cho:

2 2

' ' x y x x d = =

khi dng tnh cht hnh bnh hnh ta c :2

2 2 2' '0 ' ' 4 2 ' '

2

4 4 0

x y x y x x x y x

d d

+ = + +

+ =

' 'y x =

b) Nu ' A v ( ')x A th x l phn t duy nht ca A c

nh ngha trong a) v vi bt ky A c :

2

2 2 2

' ' , ' '

' ' '

y x x x y x x x y

x x x y x x

= + +

= +

du = t c khi v ch khi 'y x= .

Ngc li, nu ' A v ( ')x A th x khng l phn t ca A v c

phn t x :


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2'' '

ayx x

y= +

vi x gn x hn x, vi y l phn t bt k ca A sao cho:

'', 0 x x y< > v ',a x x y= < >

Tht vy,

2'' ' ' '', ' ' ''x x x x x x x x x x = < + + >

2

2' ' ' ''2 2 ,a x x x x x x

y= + < >

22 2

2' 'a

x x x xy

=

1.1.6 TNH CHT CA PHP CHIU

i) ( ) A A A

P x y P x P y + = + .

ii) ( )22 2

A AP x I P x= +

trong I l php ng nht.

iii) x H tn ti duy nht mt biu din:

( )A Ax P x I P x= +

A i P x A ; ( )A I P x A

iv) A n AP x P x khi v ch khi 0nx x

v) x A khi v ch khiA

P x x= .

vi) x A nu v ch nu 0A

P x = .

vii) 1 2A A nu v ch nu 1 2 1 , A A AP P x P x x H = .

1.1.7 PHP XP X TUYN TNH TRONG L2


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Gi s X1, X2 v Y l nhng bin ngu nhin trong L2, nu ch c th quan

st c X1, X

2m ta c lng gi tr ca Y bng cch dng t hp tuyn

tnh: 1 1 2 2'Y X X = + , 1 2, R sao cho sai st M di y c trung bnh

bnh phng t gi tr nh nht, ngha l sao cho:

( )22

1 1 2 2: 'M E Y Y E Y X X = = + =2

1 1 2 2Y X X min

Ta c th vit :

2 2 2 2 21 1 2 2 1 1 2 2 1 2 1 22 ( ) 2 ( ) ( ) M EY EX EX E YX E YX E X X = + + + .

Ly o hm ring ca M ln lt i vi 1 , 2 , dn n h phng trnhcho nghim ti u 1 2,

21 1 2 1 2 1

21 2 1 2 2 2

( ) ( ) ( )

( ) ( ) ( )

E X E X X E YX

E X X E X E YX

+ =

+ =(1.4)

Ngoi ra, ta c th dng nh l hnh chiu trong khng gian Hilbert L2 .

Ta t vn tm phn t Y trong tp ng A :

{ }2

1 1 2 2: \ :A X L X a X a X = = + vi 1 2,a a R ,sao cho :

' inf X A

Y Y X Y

= vi X A .

Nh vy, theo nh l chiu trong khng gian Hilbert 'Y A v Y tha

iu kin trn khi v ch khi 'Y A v 'Y Y A v do

1 1 2 2 , 0Y X X X < > = ,

tc l : 1 1 2 2 1

1 1 2 2 2

, 0

, 0

Y X X X

Y X X X

< > =

< > =

p dng tnh cht ca tch v hng nh ngha trn ta suy ra (1.4).

1.1.8 PHNG TRNH DON


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Cho khng gian Hilbert 2L , mt t p con ng 2A L v mt phn t

2

X L , nh l chiu trong khng gian Hilbert khng nh rng tn ti duy nhtmt phn t 'X A sao cho:

', 0, X X Y Y A< > = (1.5)

Phng trnh (1.5 ) gi l phng trnh don v phn t ' :A

X P X = l d

on tt nht ca X trong A. Hay ta c th ni don tt nht ca X trong A l

chiu ca X trong A.

1.1.9 K VNG C IU KIN V DON TT NHT TRONG L2

Nh ta ni trn, nu 2nX L ,2X L th

2L

nX X khi v ch khi:

2 20n n X X E X X = khi n

Mt s tnh cht ca shi t theo ngha bnh phng trung bnhNu

2L

nX X th khi n

i)2

nEX ,1 ,1 EXLnX X= < > < > =

ii)22 2

, ,Ln n n E X X X X X E X =< > < > =

iii) ( )2

, , , ,Ln n n nE X Y X Y X Y E X Y = < > < > = < >

nh ngha 1: ( Don bnh phng trung bnh tt nht ca Y)Nu A l mt khng gian con ng ca 2L th don bnh phng tt

nht ca Y trong A c nh ngha l phn t 'Y A sao cho :

2 2 2'

Z A Z A: inf inf Y Y Y Z E Y Z

= =

nh ngha 2: ( K vng c iu kin AE X )


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Nu A l mt khng gian con ng trong L2 v cha cc hm hng, nu

2

X L th ta nh ngha k vng c iu kin ca X vi A cho trc l phpchiu A AE X P X =

Mt khc, v ton t AE X l ton t chiu trn L2 nn

AE c cc tnh cht

php chiu :

i) ( ) , , A A AE aX bY a E X b E Y a b R+ = +

ii)2

L

A n A E X E X nu

2L

nX X

iii) ( )1 2 1 A A AE E X E X = nu 1 2A A=

1.11.21.31.41.51.61.71.8


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1.2 KHAI TRIN CHNH TC CA QU TRNH NGU NHIN

1.2.1 QU TRNH NGU NHIN BIU DIN DI DNG TNG CC

HM NGU NHIN CBN

nh ngha ( Hm ngu nhin cbn )Hm ngu nhin cbn l hm c dng :

( ) ( ).t C t = (1.6)

trong :

C l mt i lng ngu nhin

( )t l hm khng ngu nhin ca bin s t T

Cc c trng ca hm ngu nhin cbni) K vng : ( ) ( ) ( ). .C C E t E t t E = =

trong :

CE l k vng ca i lng ngu nhin C

* Nu 0CE = th ( ) 0E t =

* Khi xt cc hm ngu nhin cbn c k vng bng khng , ta k hiu l ( )0

t

=> ( )0

0E t =

ii) Hm t tng quan ca hm ngu nhin cbn ( )t :( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2' ' ' ', . . . . . CK t t E t t t t E C t t D

= = =

trong :

CD l phng sai ca i lng ngu nhin C

iii) i vi cc hm ngu nhin cbn, ta c cc php bin i tuyn tnh+ Php ton o hm : ( ) ( )

'' .t C t =


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+ Php ton tch phn xc nh : ( ) ( )

0 0

.T T

t dt C t dt =

iv) Nu G l mt ton t tuyn tnh , ta c :( ){ } ( ){ }G t C G t =

nh ngha ( Qu trnh ngu nhin theo cc hm cbn)Cho qu trnh ngu nhin :

( ) ( ) ( )1

.n

i i

i

t E t C t =

= + (1.7)

trong : iC l cc i lng ngu nhin c k vng bng 0, 1,i n=

( )E t l k vng ca ( )t .

Biu thc (1.7) c gi l khai trin ca qu trnh ngu nhin ( )t theo cc

hm cbn.vi : + cc i lng ngu nhin ( )iC t , 1,i n= c gi l h s khai trin.

+ cc hm khng ngu nhin ( )i t , 1,i n= c gi l cc hm ta .

c trng ca qu trnh ngu nhin theo cc hm cbnGi s ( )t biu din c di dng (1.7) , khi :

Xt mt ton t tuyn tnh G tc ng ln ( )t , ta s c :

` ( ) ( ){ } ( ){ } ( ){ }1

n

i i

i

t G t G E t C G t =

= = +

t ( ){ } ( )GG E t E t = v ( ){ } ( )i iG t t =

Khi :

( ) ( ) ( )1

n

G i i

i

t E t C t =

= +


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Ta thu c ( )t theo cc hm cbn vi cc h s 1 2, ,...., nC C C .

Nh vy, nu qu trnh ngu nhin ( )t khai trin di dng tng cchm cbn, qua php bin i tuyn tnh G th cc h s khai trin khng thay

i, cn k vng v cc hm ta b tc ng theo php bin i tuyn tnh.

1.2.2 KHAI TRIN CHNH TC CA QU TRNH NGU NHIN

Gi s qu trnh ngu nhin khai trin di dng :

( ) ( ) ( )1 .

n

i i

it E t C t

== + ,

trong : , 1,iC i n= l cc i lng ngu nhin c k vng bng 0 v ma trn

tng quan i jk .

Xt hm t tng quan v phng sai ca ( )t

( ) ( ) ( )0 0

' ', , K t t E t t

=

trong :

( ) ( )0

1

n

i i

i

t C t =

=

( ) ( )0

' '

1

n

i i

i

t C t =

=

Khi :

( ) ( ) ( ) ( ) ( ) ( )' ' '11

, . ,n

i j i j i j i j

i i jj

K t t E C C t t E C C t t ==

= =

vi :

( ) [ ]2

i i i iE C C E C D= = ( iD c gi l phng sai ca iC )

( ) ( ), , , 1,i j ijE C C k i j i j n= =


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Nh vy :

( ) ( ) ( ) ( ) ( )' ' '1, .n

i i i i j i j

i i j

K t t t t D t t k =

= + (1.8)

t t = t ta c phng sai ca ( )t :

( ) ( ) ( ) ( )2

1

n

i i i j i j

i i j

D t t D t t k =

= + (1.9)

* Ch :

Nu cc h s iC ( )1,i n= khng tng quan vi nhau , ngha l i jk = 0 ( i j ) .Khi ta ni (1.7) l khai trin chnh tc ca hm ngu nhin ( )t

Nhn xt* Khai trin chnh tc ca qu trnh ngu nhin ( )t l khai trinc dng :

( ) ( ) ( )1

.n

i i

i

t E t C t =

= +

trong :( )E t l k vng ca qu trnh ngu nhin ( )t

( )( )1,i t i n = l cc hm ta

( )1,iC i n= l cc i lng ngu nhin khng tng quan vi nhau v u

c k vng bng 0

* Nu ( )t c khai trin chnh tc th hm t tng quan ca n c dng l

( ) ( ) ( )' '1

,n

i i i

i

K t t t t D =

=

* Nu ( )t c khai trin chnh tc th phng sai ca ( )t c dng l :

( ) ( )( )2

1

n

i i

i

D t t D =

=


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1.2.3 A QU TRNH NGU NHIN V DNG CHNH TC

Cho qu trnh ngu nhin ( )t biu din di dng :

( ) ( )1

.n

i i

i

t M t =

= (1.10)

trong :

( )

( )1,i t i n = l cc hm khng ngu nhin

iM l cc i lng ngu nhin tng quan c ma trn tng quan :

1 12 1

2 2

.... ....

....

....

n

n

M

n

D k k

D kK

D

=

vi :

( )( ) 0, , 1, ,i j i i j jk E M E M E i j n i j = =

v 0i i EM E =

Biu thc dng (1.10) ca ( )t cha phi l dng chnh tc , do ta cn a n

v dng chnh tc.

Ta vit biu thc(1.10) di dng :

( ) ( ) ( ) ( )1 1

n n

i i i i ii i

t E t M E t = == +

t :0

i i i M M E = , ( ) ( ).i i E t E t = , 1,i n=

Khi :

( ) ( ) ( )0

1

n

i i

i

t E t M t =

= +

Biu thc trn cn c th vit di dng :


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( ) ( )* *

.T

t M t = (1.11)

vi : ( ) ( ) ( )*

t t E t = v ( )*

,M t l cc ma trn ct v T biu din php

chuyn v ca ma trn

Ma trn tng quan c vit di dng :

**T

MK E M M

=

Chn ma trn A sao cho vect:

*

.C A M= c cc thnh phn iC , 1,i n= l cc ilng ngu nhin khng tng quan

* ** *

. T T T T T C

TM C

K E C C E A M M A AE M M A

A K A D

= = =

= =

(1.12)

vi :

CD l ma trn ng cho m cc phn t trn ng cho l phng sai

ca iC , 1,i n=

1

2

0 ... 0

0 ... 0

... .... ... ...

0 0 0n

C

C

C

C

D

DD

D

=

Biu thc ( 1.12) ta thy ma trn A chuyn ma trn tng quan CK v dng

ng cho

Ma trn MK l i xng v thc , v vy tn ti ma trn trc giao A tha :

i j n nA a =

Ta c :


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( ) ( ) ( ) ( ) ( ) ( )** * *

1 1T T

TT Tt M t A A M t A M A t C t

= = = =

vi :*

C A M=

( ) ( ) ( ) ( ) ( ) ( )1T T

Tt A t A t A t = = =

(do A l ma trn trc giao nn 1 TA A = )

Nh vy ta c qu trnh ngu nhin c a v dng chnh tc :

( ) ( ) ( )1

.

n

i i

it E t C t

== +

1.2.4 MT S KHAI TRIN CHNH TC C BIT Khai trin Karhunen Love

Qu trnh Wiener ( ){ }W t ,0 1t khai trin theo cng thc :

( ) ( ) ( )t 0W i ii X t

== trong : iX l cc i lng ngu nhin c lp c phn phi chun :

iEX 0= ,

( )2

1, 0,1,2,...,

2 12

i DX i n

i

= =

+

( )i t l cc hm khng ngu nhin xc nh bi :

( ) ( )2 sin 2 1 , 0,1,2....2i

t i t i

= + =

* Dy hm ( ){ }i t c th xem nh mt h trc chun y trong [ ]2 0,1L

vi :


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( ) ( ) ( )1

0

, 0i j i jt t dt = = , i j

( ) ( )0

min( , )i ji

s t s t

=

=

* Mt khc, dy hm ( ){ }i t c th xem nh hm ring ca ton t B c

xc nh bi cng thc:

( ) ( ) ( )1

0

, B t B s t t ds =

vi ( ), min( , ) B s t s t =

Cc gi tr ring ca ton t B l :

( )2

2 12i

i

= + , i = 0, 1, 2 .

Khai trin theo cc hm SchauderXc nh cc hm Haar bi cc biu thc sau :

( )1 1 0 1 I t t =

( )2

11 0

21

1 12

khi t

I t

khi t


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( )

( )

12

122 1

2 0 2

2 2 2

0 2 1

n

nn

nn n

n

khi t

I khi t

khi t

+

+ +


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1.3 C S TRC GIAO V TRC CHUN TRONG KHNGGIAN HILBERT

1.3.1 NH NGHA ( Trc giao, trc chun )

Hai phn t x, y ca khng gian Hilbert c gi l trc giao, y nu:

, 0x y = .

Cho tp hp S H ta vit x S nu mi y S , x y . Phn b trc giao cho

tp S trong H, k hiu S , l tp tt cx H sao cho x S .

Tch v hng trn khng gian Hilbert H c xc nh:

, x x x=

Tp hp hm { }k k K trong khng gian H l tp trc giao nu mi phn t l

trc giao. Chng hn, , 0, , ,i j

i j i j K N =

Nu hm c chun ha sao cho 1,k k = khi tp hp c gi l tp

trc chun.

1.3.2 NH NGHA (Cs)

Nu tp hp { }k k K bao gm cc hm c lp tuyn tnh trong H c

gi l csca H. S nhng phn t csc gi l chiu ca H.

Cho c s { }k k KB = ca khng gian Hilbert H, tn ti vi bt k

x H mt tp duy nht h s{ }k k Kx sao cho :

0k k

k K

x x

= (1.13)

H s ny c gi l h s khai trin ca x i vi csB. T (1.13 )

vit n gin l:


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k kk Kx x

=

(1.14)

1.3.3 NH NGHA (Cstrc giao v trc chun)

Cstrc giao ca khng gian Hilbert H l tp trc giao { }k k K trong

H. Nu trong hm cng tnhk

l chun, chng hn 1,k k = , khi n l c

strc chun.

Tnh cht ca c s trc chun l khai trin h s kx bi tch v hng

ca x vi hm cstrc chun k :

,k kx x = (1.15)

Bt k tp hp ca hm c l p trong khng gian Hilbert H c th bin

i thnh tp trc chun.

Hin nhin bt k tp trc giao c th thnh trc chun do c chun ha

n gin. Do , hu nh ta xt trc chun hn tp trc giao v cskhng mttnh tng qut.

Xt khng gian con S ca khng gian Hilbert H. Khi ta xc nh php

chiu trc giao trn S nh sau:

1.3.4 NH NGHA ( Php chiu trc giao )

Xt cs trc chun { }k k K

ca khng gian con S ca H. Php chiu

trc giao ca x H trn S, k hiu: SP x , c cho:

,S k k

k K

P x x

= < >


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1.4 QU TRNH WIENER

Qu trnh Wiener l mt v d rt quan trng i vi l thuyt xc sut

thng k. Qua th nghim ca Brown qu trnh ngu nhin ny c dng lm

m hnh chuyn ng ca ht di tc ng va chm hn lon ca cc phn t .

1.4.1 NH NGHA ( Qu trnh Wiener )

Ta gi qu trnh ( )W t l qu trnh Wiener tha mn cc iu kin sau:

i) W(0)=0 ( h.c )

ii) Vi mi 0 10 ... nt t t < < < cc i lng ngu nhin

( ) ( )1 0 2 1 1( ) , ( ),..., ( ) ( )n nW t W t W t W t W t W t l i lng c lp

iii) W(t) c phn phi chun vi k vng bng 0 v phng sai t

( ){ }

2

2W(t) a,b12

b u

t

aP e dut

=

iv) ( )W t l qu trnh lin tc, tc hu ht cc quo ca ( )W t l hm

lin tc.

* Mt qu trnh Wiener ( )W t vi tham s phng sai bng 1 c gi l

qu trnh Wiener tiu chun ( hay chuyn ng Brown tiu chun ).

* Nu ta thay W(0) = 0 bi W(0)=x ta s c qu trnh Wiener xut pht

t x

1.4.2 CC TNH CHT QU TRNH WIENER V O WIENER

Tnh cht 1:Nu ( )W t l qu trnh Wiener khi :

( ) ( ){ }W t W h E t h=


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trong : min ( , )t h t h =

Chng minh:

Gi s, 0h t , ta s c :

( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

2

E W t W h E W h W t W h W h

E W h E W t W h W h

h E W t W h E W h

h h t

= +

= +

= + = =

Do ( )W h c phn phi chun ( )0,N h v ( ) ( )W t W h l c lp vi ( )W h .

Tnh cht 2 :Cho [ ], X a b= v [ ],Y c d= , trong a b< v c d<

xc nh :

W(X)= W(b) - W(a)

W(Y)=W(d)-W(c) Khi :

{ } ( )W(X) W(Y) E X Y =

trong : l o Lebesgue

Chng minh:

Khng mt tnh tng qut, gi s vi a c , khi :

0

( )

khi b c

X Y b c khi c b d

d c khi d b

= <


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( ) ( ){ }

0

W X W Y

khi b c

E b c khi c b d

d c khi d b

= <


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Vy (1.17) tha mn.

Bng phng php quy n p ta gi thit :

( ) ( )1k-1

j jj=1 , 1

1exp i W t exp -

2

k

j l j l

j l

E t t

=

=

(1.18)

vi mi s k, ta ch ra rng tnh cht ny ng vi n k=

Ta xt :

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

1

j j k 1 1

1

j k -1 k -1 k 1

1'

k k -11

W t W t W t

W t W t W t W t

W t W t W

k k

j j k

j j

k

j k k k

j

k

k j j

j

i i i

i i i i

i i t

= =

=

=

= +

= + +

= +

trong : 'j j = vi 1,2,..., 2j k= v'

1 1k k k = + .

Nh vy:

( )j1

exp i W tk jj

E =

= ( ) ( )( ) ( )k -1

'k k -1 j j

j=1

exp i W t exp i W tkW tE

Do tnh cht s gia c lp ca ( )W t , khi 1j jt t + ta c ( ) ( )k k -1W t W t c

lp vi mi ( )jW t vi 1j k

Bi vy :

( )k

jj=1

exp i W tjE

= ( ) ( )( ){ } ( )k -1

'k k k -1 j j

j=1

exp i W t W t exp i W tE E


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= ( ) ( )( ){ } ( )1

' 'k k k -1

, 1

1exp i W t W t exp -

2

k

j l j l

j l

E t t

=

do gi thit (1.18).

Khi ( ) ( )k 1W t W tk c cng phn phi vi ( )k 1W t kt

Ta c :

( ) ( )( ){ }k k k -1exp i W t W tE = ( )( ){ }k k 1exp i W t kE t

=

( )

2

1

1exp -

2 k k kt t

t cstrn ta c cng thc (1.17)

Nh vy ta c:

( )k

j jj=1

exp i W tE

= ( ) ( )1

2 ' '1

, 1

1 1exp -

2 2

k

k k k j l j l

j l

t t t t

=

(1.19)

m :

( ) ( )'1

2 '1

, 1

1 12 2 j j

k

k k k l l i l

t t t t

=

+

= ( )' '2 2 2' ' ' 2 2

1 1 1 1, 1 1

1 1 1 1( )

2 2 2 2 j j j jk k

l l k k k k k k k j l j

t t t t t t

= =

+ + +

=, 1

1( )

2 j jk

l lj l

t t =

(1.20)

thay (1.20) vo (1.19) ta c:

( )k

j jj=1

exp i W tE

= ( )1

exp -2 j l j l

t t

.

Bng php quy np ta suy ra iu phi chng minh.

o Wiener :Cho 1 20 ... nt t t< < < < v xc nh :


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( ) ( ) ( ){ }1 1 2 2: , ,..., n n A t B t B t B = .

trong : iB l tp hp Borel ca ng thng thc vi 1,2,...,i n=

Khi :

1 2 1 1

1 1 1( ) ...

2 2 ( ) 2 ( )n nA

t t t t t

=

1 2

1 2

22 2n 11 2 1

1 2 1 1

...(x )x (x )

... exp - exp - ... exp -2 t 2( ) 2( )

n

n

B

n

n nB B

dx dx dxxx

t t t t

Chng minh :

Ta c:

{ }1 1 2 2( ) ( ) , ( ) ,..., ( )n n A P t B t B t B =

( ){ ( ) ( ) ( )1 1 2 1 1 2W t , W t W t W t ,..., P B B= +

( ) ( ) ( ) }n n -1 n -1W t W t W t nB +

( ) ( ) ( ) ( ){1

1 1 2 1 1 2, ,...,B

W t dx W t W t W t BP + =

( ) ( ) ( ) }n n - 1 n -1W t W t W t nB +

( ){ ( ) ( )1

1 2

1 1 2 1 2... W t , W t W t ,...,n B B B

P dx x dx + =

( )( )

}1 1n n n n

W t W t x dx +

.

Hn na, ( ) ( )1i iW t W t c cng phn phi vi ( )1i iW t t .

Nh vy :

( ) ( ){ ( )1 2

1 1 2 1 2... W t , W t ,...,n B B B

A P dx t dx = ( ) }n 1 1W t n n nt x dx +

( )xW t k hiu l qu trnh Wiener ti x, khi :


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( ) ( ){ ( )1

1 2

1 1 x 2 1 2... W t , W ,...,

n B B B

A P dx t t dx = ( ) }n-1x n 1W t n nt dx

Khi , do tnh cht ca s gia c lp, ta c :

( ) ( ){ } ( ){ }1

1 2

1 1 x 2 1 2... W t W ,....,n B B B

A P dx P t t dx =

( ){ }n -1x n 1W t n n P t dx Nh vy :

( )( ) ( )1 2 1 1

1 1 1...2 2 2 n n

At t t t t

=

( )( )

( )( )

1 2

22212 11

1 21 2 1 1

... exp .exp ... ...2 2 2

n

n n

n

B B B n n

x xx xxdx dx dx

t t t t t

Tnh cht 4 :Tng bnh phng cc gia s ca qu trnh Wienerng vi phn hoch

0 1 ... na t t t b= < < < = ca on t a n b hi tn b a theo bnh phng trung

bnh khi lm mn phn hoch :

( )( )1

i+1

1 2

ax t 00

limi i

i

n

t tm t

i

X X b a+

=

=

Chng minh:

Ta c :

( ) ( )

( ) ( )

1 1

1

21 12

0 0

11

10 0

i i i i

i i

n n

t t t t

i i

nn

t t i i

i i

E X X E X X

D X X t t b a

+ +

+

= =

+= =

=

= = =

Do tnh c lp ca1i it t

X X+

( i = 0,1,, n 1 )


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( ) ( )1 11 12 2

0 0

i i i i

n n

t t t t

i i

D X X D X X + +

= =

=

( ) ( )1 121 4 2

0i i i i

n

t t t t

i

E X X E X X + +

=

=

( ) ( ) ( )1 1

2 2 2

1 1 10 0

3 2n n

i i i i i i

i i

t t t t t t

+ + += =

= =

( ) ( ) ( ) ( )1

i +1 1 i +10

2 ax t 2 ax tn

i i i i

i

m t t t b a m t

+=

=

Vy : khi ( )i +1ax t 0im t th ( )11 2

0

0i i

n

t t

i

D X X +

=

T :

( ) ( ) ( )1 12

1 12 2

0 0

0i i i i

n n

t t t t

i i

E X X b a D X X + +

= =

=

khi

( )i +1ax t 0im t

hay ( ) ( )11 2

0i i

n

t t

i

E X X b a+

=

khi lm mn phn hoch

Tnh cht 5:Cho W(t) l qu trnh Wiener tiu chun, khi qu trnh :

1

( ) W( ) ; 0,t(0) 0,

K t t t

K

= >=

cng s l qu trnh Wiener tiu chun.

Chng minh:

chng minh tnh cht ny ta dng phng php hm c trng

Khi t > u ta xt :


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( ) ( )( ){ } 1 1exp i exp i tW Wt u

E K t K u E u =

= ( )1 1 1

exp i W W Wt u t

E t u u

= ( )2 2 2

2 1 1 1exp

2 2

ut u

t u t

= ( )2

exp -2

t u

(1.21)

Nh vy ( ) ( )K t K u c phn phi chun N (0, t -u ).

Tnh c lp ca cc s gia ca qu trnh K(u) , c suy ra t h thc sau:

( )( ) ( ) ( ) ( )( )( ) (0) ( ) (E K u K K t K u E K u K t K u =

=1 1 1

W W Wt u

E u t uu

= 2 21 1 1W W Wu

E u t ut u

= u u = 0 (1.22)

T (1.21) v (1.22) ta suy ra K(u) l mt qu trnh Wiener tiu chun.

Tnh cht 6Cc quo ca qu trnh Wiener hu ht khng u kh vi, cho d chng

lin tc hu chc chn :

P { ( ): tW l kh vi } = 0

Tnh cht 7 :Hu chc chn hm ( )W t khng c bin phn b chn trn bt k khong

hu hn no :


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( ) ( )1

1 i0

: W W t 1 0n

i

i

P t A A

+=

> >

Tnh cht 8:W(t) tun theo lut lga lp, ngha l :

( )

( )

W tlimsup 1 1

2tlnlnt

W tliminf 1 1

2tlnlnt

t

t

P

P

= =

= =

Lut lga lp a phng ca qu trnh Wiener :

( )

( )

0

0

W tlimsup 1 1

12tlnln

W tliminf 1 1

12tlnln

t

t

P

t

P

t

= =

= =

Tnh cht 9:( )W t l mt Mactingan i vi WtF

Tnh cht 10 : (c trng Levy ca qu trnh Wiener )( )W t l qu trnh Wiener khi v ch khi :

+ ( )W t l mt mactingan , ( )W 0 0= hu chc chn.

+ ( )2

W t t l mt mactingan ( i vi WtF )


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1.4.3 QU TRNH WIENER n - CHIU

Mt qu trnh Wiener n - chiu l vect ( )W t sao cho :

( ) ( ) ( ) ( )( )1 2 nW t W , W ,..., Wt t t=

trong : ( )iW t l qu trnh Wiener mt chiu v, vi i j , qu trnh ( )iW t v

( )jW t c lp.

Tnh cht 1:Nu ( )W t l qu trnh Wiener n - chiu :

( ) ( ){ } ( )i j ,W W i j E h t h t =

Chng minh :

R rng nu i j khi ( )iW h ) v ( )jW t c lp v bi vy :

( ) ( ){ } ( ){ } ( ){ }i j i jW W W W 0 E h t E h E t = =

Hn na, nu i j= ta d dng c tnh cht 1. Tnh cht 2 :

Vi 1 2( , ,..., ) j j jnn

j R = v 0jt trong 1,2,...,j m= :

( ) ( )1 , 1

1exp . exp .

2

m m

j j j k j k

j j k

i W t t t E = =

=

Chng minh :Theo tnh cht 3 trn , khi s dng tnh c lp ca thnh phn qu trnh

Wiener ( )W t ta c :

( ) ( )m m

j j k j=1 1 1

exp i .W t exp i Wn

jk j

j k

E E t = =

=


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=( )

m

jk k j

j=11

exp i .W tn

k

E

=

( ), 11

1exp -

2

n m

j k l k j l

j lk

t t ==

=

( ), 1 1

1exp -

2

m n

jk lk j l

j l k

t t = =

=

( ), 1

1exp - 2

m

j l j l

j l

t t =

=


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1.5 TCH PHN ITO

1.5.1 NH NGHA

Gi s trn khng gian ( ), ,F P cho h hm tng cc - i s

tF F , 0t v qu trnh Wiener ( )W t , 0t , W(0) = 0 vi quo lin tc

tng thch vi h tF sao cho s gia ( ) ( )W h W t sau thi im t c lp vi

- i s tF .Cho T l mt s khng m , ta xt TN l lp cc hm ngu nhin

[ ]: 0, f T R tha cc iu kin sau :

i) ( ) 2

0

T

f t dt E <

ii) f( t ) l hm o c

iii) ft l tng thch i vi tF , ngha l tf l tF o c xy dng khi nim tch phn Ito ca hm ngu nhin thuc lp TN

trc ht ta xt cc hm scp, ngha l hm c dng :

1

1

0 ,( ) ( ) ( )

k k

n

kk

t tf t f t t

+

=

= (1.23)

trong : 0 1 20 ... nt t t t T = < < < < = l mt phn hoch ca [ ]0,T ,

f (tk) l cc bin ngu nhin o c i vi tF , vi k = 0, 1, 2, , n ;I l hm ch tiu

Vi cc hm scp c dng (1.23) ta xc nh tch phn Ito ca hm ngu nhin

cbn bi :

( ) ( )( )kn -1

1k = 00

t( ) W(t) = f ( ) W t WT

k k f t d t +

ViT

f N s tn ti dy hm sc p ( )n

f t b chn sao cho :


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( ) ( )( )2

0lim 0

T

nn

E f t f t dt

=

T ta nh ngha tch phn Ito cho hm ngu nhin thuc lp TN theo

h thc sau

0 0( ) W(t) : lim ( ) W(t)

T T

nn

f t d f t d

=

1.5.2 CC TNH CHT CBN CA TCH PHN ITO

Tnh cht 1: ( tuyn tnh )( )

T

0 0 0

( ) ( ) ( )dW(t) + g (t)ddW(t) W(t)T T

f t g t f t + = ,

trong : , : l nhng hng s

Chng minh:

Ta gi s :

)11

,0

( ) ( ) ( )k k

n

k t tk

f t f t t+

=

=

)1

1

,0

( ) ( ) ( )k k

n

k t tk

g t g t t+

=

=

Do nh ngha

( )

0

( ) ( ) W(t)T

f t g t d + = ( )( )1

1 k0

( ) ( ) W ( ) W (t )n

k k k

k

f t g t t

+=

+

= ( ) ( ) ( )( ) ( ) ( )( )1 1

k +1 k k +1 k 0 0

W t W t ( ) W t W tn n

k k

k k

f t g t

= =

+

=0 0

( ) W(t) ( ) W(t)T T

f t d g t d +

Tnh cht 2:


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0

( ) W(t) 0T

f t d E

=

Chng minh:

( )kf t v ( ) ( )k+1 k W t W t c lp vi nhau

Bi vy:

( ) ( ) ( ){ }1 kW tk k f t W t E + = ( ){ } ( ) ( ){ }k +1 k W t W tk f t E E

Khi theo nh ngha :

0

( ) W(t)T

f t d E

= ( ) ( ) ( ){ }1

k 11

W t Wn

k kk

E f t t

+=

= ( ){ } ( ) ( ){ }1

k 11

W t Wn

k k

k

E f t E t

+=

= 0

Tnh cht 3: (ng cIto )( ) ( ) ( ) ( ) ( ) ( )

0 0 0

W t W tT T T

E f t d g t d E f t g t dt

=

Khi ( ) ( ) f t g t ta s c :

( ) ( ) ( )2

2

0 0

W tT T

E f t d E f t dt

=

Chng minh:

Do nh ngha

( ) ( ) ( ) ( )0 0

W WT T

f t d t g t d t E


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= ( ) ( ) ( ) ( ) ( ) ( )1

i +1 i k +1 k

, 0

W t W t W t W tn

i k

i k

f t g t E

=

Nu i < k khi do tnh cht c lp

Ta c :

( ) ( ) ( ) ( ) ( ) ( ){ }i +1 i k +1 k W t W t W t W ti k f t g t E

= ( ) ( ) ( ){ } ( ) ( ) ( ){ }i +1 i k +1 k W t W t W t W ti k f t E g t E

= 0

Nh vy,

T

0 0

( ) W(t) g (t) dW(t)T

f t d E

= ( ) ( ) ( ) ( )1 2

k +1 k 0

W t W tn

k k

k

f t g t E

=

= ( ) ( ){ } ( ) ( ){ }1 2

k +1 k 0

W t W tn

k k

k

f t g t E E

=

= ( ) ( ){ } ( )

{ }

1 2

k +10

W tn

k k k

k

f t g t E t E

=

= ( ) ( ){ }( )1

10

n

k k k k

k

E f t g t t t

+=

= { }0

( ) ( )T

E f t g t dt

Tnh cht 4 :

0

( ) W(t)T

f t d l hm o c i vi tF

Chng minh:

Khi ( ) ( ) ( )( )1

k +1 k 00

( ) W t W tW(t)T n

k

k

f t d f t

=

=

V do nh ngha ( )f s v ( )W s o c i vi tF , nn:


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T

0

( ) W(t) f t d o c i vi tF

Tnh cht 5( )

0

( ) W(t)T

X T f t d = l mt martingale i vi tF

Chng minh :

Vi s T< , do tnh cht 2 , tnh cht 4 v tnh cht s gia c lp

Ta c :

( ){ } ( ) ( ) ( ){ }

( ) ( ) ( ){ } ( )

X T \ \s sE F X s E X T X s F

X s E X T X s X s

= +

= + =

Tnh cht 6 :Vi 0 s u T <

t( ) W ( )dWt ( ) WtT u T

s s u

f t d f t f t d = +

Tnh cht 7:[ ] ( ) ( )1 2 t 2 1,

0

. W . W t W tT

t t f I d f = .

trong : f l bin ngu nhin ty o c i vi ( )1 1 2t F t t v bnh phngkh tch

1.5.3 TCH PHN ITO NHIU CHIU

Vectngu nhin

( ) ( )1 nW t W ,...., Wt t=


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c gi l qu trnh Wiener n - chiu nu :

+ Mi thnh phn

k

tW ( 1,2,... )k n= l qu trnh Wiener 1 chiu+ Cc thnh phn ktW , 1,...,k n= l nhng qu trnh ngu nhin c lp

Gi s ( )ija a t = l ma trn m n sao cho mi ij( )a t thuc TN .

Khi ta nh ngha : ( )0

W tT

X a d =

l ma trn 1m ( hay vectct m - chiu ) m thnh phn th i ca n l :

jij t

1 0

W , 1,...,Tn

i

j

X a d i m=

= =

1.5.4 VI PHN NGU NHIN CA HM HP, CNG THC ITO

Trong php tnh vi phn thng thng c vi phn ca o hm hp nhsau :

Gi s tX l hm kh vi sao cho :

( )dX t a dt =

Gi s g (t, x) l hm hai bin kh vi. Khi cng thc tnh vi phn ca

hm s hp ( )( ),Y g t X t = dng :

( ) ( )( ) ( )( ) ( )

( )( )

( )( )

, ,

, ,

g gdY t t X t dt t X t dX t

t t

g gt X t t X t a dt

t x

= +

= +

Ta xt vi phn ngu nhin Ito , ta c :

Cng thc Ito 1- chiu : nh ngha:

Tch phn ngu nhin Ito hay qu trnh Ito (1chiu ) l qu trnh ngu

nhin lin tc ( )X t trong 2L c dng :


44/77



( ) ( ) ( ) WdX t a t dt b t d t = +

nu:0 0

( ) (0) ( ) ( ) Ws ,0t t

X t X a s ds b s d t T = + +

trong : ( )a t v ( )b t l cc qu trnh ngu nhin tng thch i vi ( )F t o

c sao cho :

( )

( )

0

2

0

1

1

T

T

P a t dt

P b t dt

< =


45/77



nh ngha:Tch phn ngu nhin Ito hay qu trnh Ito n - chiu l qu trnh ngu

nhin vect lin tc ( ) ( )1( ),..., ( )nX t X t X t = sao cho mi thnh phn ca n l

qu trnh Ito:

i10 0

( ) (0) ( ) W ( ), 0t t n

i i i ij

j

X t X a t dt b d t t T =

= + +

trong : { }( ) , 1,...,ia t i n= v { }( ) , , 1,...,ijb t i j n= l cc qu trnh ngu nhin

tng thch i vi tF sao cho :

0

2

0

( ) 1

( ) 1

T

i

T

ij

P a t dt

P b t dt

< =


46/77



v ( )B t l ma trn cc hm o c ( )jib t 1,2,...,i k= 1,2,...,j m=

Ta c th vit di dng ta nh sau :

1 1 1 1 1 m1 t t

2 2 2 2 2 m1 t t

k m1 t t

W ... W

W ... W

...............................

W ... W

t m

t m

k k k k

t m

dX a dt b d b d

dX a dt b d b d

dX a dt b d b d

= + + +

= + + + = + + +

Cng thc vi phn Ito mrng c dng :Vi qu trnh ngu nhin k- chiu , c vi phn ngu nhin (1.21) v Y(t) l

hm kh vi lin tc mt ln theo t , hai ln theo xi v xjng thi cc o hm

ringix

Y

b chn

Ta c:

jii

1 1

( ) ( ). ( ) Wx

n ki

j

j i

YdY t t b t d

= =

= +

2

i1 , 1 1

1( ) ( ) ( ) ( ) ( ) ( )

2 x

k k ni i j

l li ji i j l

Y Y Yt t a t t b t b t dt

t x x= = =

+ +


47/77



CHNG 2

A THC HERMITE V KHAI TRIN

FOURIER HERMITE

2.1 A THC HERMITE

Trn trc thc ( ),R = + vi o Gauss :

( ) ( )dx x dx = , (2.1)

trong 2

2( )1

2

x

x e

= .

2.1.1 NH NGHATa nh ngha khng gian cc hm bnh phng kh tch vi o Gauss

l :

2 2( , ) ( ), ( ) ( ) L R f x f x dx

+

=

<

Tch v hng trong khng gian ny c xc nh l :

, ( ) ( ) ( ) ( ) ( ) ( ) f g f x g x dx f x g x x dx

+ +

< > = = Gi s, l bin ngu nhin Gauss chun vi phn phi ( )0,1N ,khi :

, = E [f ( ) g( )]f g < >

vi E l ton t k vng

Ta k hiu khng gian Hilbert trong nh ngha trn l ( )2 ,L R


48/77



2.1.2 LIN H GIA A THC TRC GIAO V A THC HERMITE

a thc Hermite c xc nh l:

( ) ( )2 2

2 21 , 0,1,2...n

n

x xn

n

d P x e e n

dx

= = (2.2)

( )nP x l a thc trc giao i vi o Gauss

( ) ( ), , ,n m n mn m N P P E P P < > =

( )! ,n n m=

=0 , n m

n! , n m

=

Do a thc Hermite chun ha c xc nh :

( )22

21

2 2( )

( !) ( 1) ( )!

x

n

xnnn

nH x

P x dn e e

dxn

= = (2.3)

( ){ }, 0,1,...n H x n= l cstrc chun trong khng gian Hilbert 2 ( , )L R .

Khi ( )0 1H x = , ta c:

( ) ( ) ( ) ( ,1) 0,n n n E H H x d x H +

= = = nu n 0 (2.4)

Bi vy a thc Hermite ca bin ngu nhin Gauss chun c k vng bng 0

Cng nh hu ht nhng a thc trc giao, a thc Hermite c hm sinh

2

2( , )z z x

x z e +

= (2.5)

Khai trin ( , )z thnh chui Taylor ca bin z ( x c xem nh mt tham s)

Ta c:

00

( , )( , ) ( )

!

n n

znn

x z zx z

z n

==

=


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Mt khc,

( )( )

22 2

2 2 2,

z xz xzx

x z e e e

+ = =

( )2 2

2 2

0

1.

!

nx xnn

nn

de e z

n dx

=

=

=( )

0 !

nn

n

P xz

n

= (2.6)

Bi vy h s khai trin Taylor ( , )x z l a thc Hermite

( ) 0( , )

n

n

n z

x zP x

z

=

=

(2.7)

Hm sinh (2.5) l hm ly tha trong tnh cht ca a thc Hermite

2.1.3 O HM CA A THC HERMITE

o hm 2 v cng thc (2.2) i vi x ta c :

( ) ( )2 2 2 2

n 1' 2 2 2 2

1d1

x x x xnn

n n nd P x x e e e e

dx dx

+

+ =

=> ( ) ( ) ( )' 1n n n P x x P x P x+= (2.8)

Mt khc, o hm 2 v (2.6) i vi x ta c :

( ) ( )( )

( )

1

0'

0

, ,!

!

n n

n

n n

n

P x x z z x z z

x n

P xz

n

+

=

=

= =

=

S chuyn ly ch s tch phn trong

( ) ( )'10 0! !

n nn n

n n

P x P xz z

n n

+

= =

=


50/77



( )

( )( )'1

1 01 ! !

n n nn

n n

P x P xz z

n n

= =

=

So snh h s nz ta c :

( ) ( )' 1n n P x n P x= ( )1n (2.9)

H thc quy ca a thc Hermite :

Ta c th s dng h thc quy o hm php ton Hermite

Kt hp cng thc (2.8) v (2.9) trn

( ) ( ) ( )

( ) ( )

'

1

'

1

n n n

n n

P x x P x P x

P x n P x

+

=

=

Ta suy ra c :

( ) ( ) ( )1 1n n n x P x P x n P x+ =

Hay

( ) ( ) ( )1 1 , 1,2,...,n n n P x x P x n P x n+ = =

T cng thc (2.8) ta c :

( ) ( ) ( )'1 , 1,2,...n n n P x x P x P x n+ = = (2.10)

o hm i vi x ta c :

( ) ( ) ( ) ( )' ' ''1n n n nP x P x x P x P x+ = +

( ) ( ) ( ) ( )' ''( 1) n n n nn P x P x x P x P x+ = +

hay ( ) ( ) ( )'' ' 0n n n P x x P x n P x + = (2.11)

y l php ton a thc Hermite

S dng h thc quy (2.10)( ) ( ) ( )1 1 , 1,2,...,n n n P x x P x n P x n+ = =


51/77



vi ( ) ( )1 00, 1 P x P x = =

Ta c th d dng c cc a thc Hermite l :( ) ( )1 0 .1 P x x P x x x= = =

( ) ( ) ( ) 22 1 0 . 1 1P x x P x P x x x x= = =

( ) ( ) ( ) ( )2 3 33 2 12 1 2 2 3P x x P x P x x x x x x x x x= = = =

( ) ( ) ( ) ( ) ( )3 2 4 2 2 4 24 3 23 3 3 1 3 3 3 6 3P x x P x P x x x x x x x x x x= = = + = +

( ) ( ) ( )

5 3

5 4 34 10 15 P x x P x P x x x x= = +

H thc quy ca a thc Hermite chun:Ta c :

( ) ( ) ( )2 2

n12 22

n

d! 1

dx

x xn

n H x n e e

=

o hm 2 v ta c :

( ) ( ) ( )2 2 2 2

n 11' 2 2 2 22

n 1

d! 1

dx

x x x xnn

n n

d H x n x e e e e

dx

+

+

= +

( ) ( ) ( )' 1 1 1n n n H x x H x H n+ = + + (2.12)

S chuyn ly ch s tch phn v so snh h s zn ta c :

( ) ( )' 1n n H x n H x= (2.13)

Vi a thc Hermite chun, kt h p (2.12) v (2.13) h thc quy trthnh

( ) ( ) ( )1 11n n n x H x n H x n H x+ + =

( ) ( ) ( )1 11 0n n nn H x x H x n H x+ + + = (2.14)

vi ( ) ( )1 00, 1 H x H x = =

Vy theo quy tc o hm ta c


52/77



( ) ( )' 1n n H x n H x=

2.1.4 CC B CA A THC HERMITE

B 1 :( ) ( ) ( ) ( )1

1

1n n

d x H x x H x

dxn + =

+, (2.15)

trong ( )x l mt Gauss (2.1)

Chng minh :

Do php tch phn v h thc quy (2.14) ta c:

[ ( ) ( )] '( ) ( ) ( ) ' ( )n n nd

x H x x H x x H xdx

= +

Theo cng thc (2.13) : ( ) ( )' 1n n H x n H x=

nn :

( ) ( ) 1( ) ( ) ( ) ( )n n nd

x H x x x H x n x H xdx

= +

= ( ) ( ) ( )1n n x H x n H x +

= ( )11 ( ) nn x H x ++

do cng thc (2.14)

Chia 2 v cho 1n + , ta c (2.15)

B 2:Vi bt k s nguyn k v l khng m , k hiu k l { }min ,k l=

Ta c:


53/77



2( ) ( ) ( , , ) ( )

k l k l h

h k l

H x H x Z k l h H x+

= (2.16)

trong :

( )1/2

2, , ( )( )( )k l k l h

h h k hZ k l h

+

=

(2.17)

Chng minh:

T cng thc (2.6) ta c :

( ) ( )( ) ( )

0 0

, ,! !

k lk l

k l

P x P x x z x z

k l

= =

= (2.18)

Mt khc:

( ) ( )2 2

2 2, , .z

zx x

x z x e e

+ +

=

=( )

2 2

2

zz x

e

+ + +

=( )

22

2 2

z xx

ze e e

+

=( ) ( )

( )0 0! !

h

vv

h v

z P xz

h v

= =

+

0 0

( ) 1( ) ( )

! !

h vv

h v

P xz z

h v

= =

= +

=

0 00

( ) 1!!v u vh

u h v h uvu

vP x zvh

= =

+ +

Cho v u s= + khi u v th tng ng 0s . Cng thc trn c th vit:

( ) ( )( )

0 0 0

, ,! ! !

u s u h s h

h u s

P x x z x z

h u s

+ + +

= = =

=


54/77



k hiu ,u h k s h l + = + =

Khi 0, 0u k h s l h= = , ta c h k l Thay i tng trn ta c:

( ) ( )( )

0 0

, ,! ! !

u s k l

k l u h k s h l

P x x z x z

h u s

+

= = + =+ =

=

( )

( ) ( )2

0 0 ! ! !

k l h k l

k l h k l

P xz

h k h l h

+

= =

=

Bi vy , ta c:

( ) ( )( )

( ) ( )2

0 0

, ,! ! !

k l h k l

k l h k l

P x x z x z

h k h l h

+

= =

=

(2.19)

So snh cng thc trn vi (2.18) ta c:

( ) ( )

( ) ( )

( )2! !

! ! !

k l k l h

h k l

k l x P x P x

h k h l h

P +

=

(2.20)

m ( ) ( ) ( )1/ 2

!n n P x n H x= nn :

( ) ( ) 2 ( ) ( , , )k l k l hh k l

H x H x H x Z k l h+

= .

B 3:Gi s Hn(x) l a thc Hermite chun v l bin ngu nhin Gauss

chun.

Khi :a

[ ( )]!

n

n E H an

+ =

trong : a l hng s ty .

Chng minh:

T hm sinh (2.6) ta c:


55/77



0

( )( , )

!n

nn P x aa z zn

=

++ =

Mt khc:

( )( )

( )

( )

( )

2 2

2 2

0 0

0 0

,

! !

! !

z z z x a x z

a z

iji j

i j

in i n

n i

a z e e e

P xaz z

i j

P xaz

i n i

+ + +

= =

= =

+ = =

=

=

So snh 2 cng thc trn ta c:

0 0 0

( ) ( )

! ! ( )!

in nn n i

n n i

P x a P xaz z

n i n i

= = =

+=

( )( )

( )0

!

! !

in

n n i

i

a na P x

i n iP

=

+ =

Ly k vng 2 v :

=> ( ) ( )0

![ ( )]

! ( )!n

in

n ii

E P aa n

E Pi n i

=

+ =

= ( )0anE P vi i = n

*** Ch : E[ Pk() ] = ,0k =0 khi k 0

1 khi k = 0

Nh vy ,ta c :

( )n

HE p + =[ ( a)]

!

nE P

n

+=

a

!

n

n


56/77



2.2 KHAI TRIN FOURIER - HERMITE CA HMBIN NGU NHIN GAUSS

2.2.1 KHAI TRIN FOURIER - HERMITE

Khi a thc Hermite l c s trc giao trong ( )2 ,L R , vi bt k hm

( )2( ) , f x L R tn ti khai trin Fourier - Hermite

0

( ) ( )n nn

f x f H x

==

trong ( ) ( ) ( )n n f x H x dxf

=

Mt khc, ta xem ( )2 , f L R nh mt hm ca bin ngu nhin n v

Gauss vi ( )2E f < + . Bi vy hm ngu nhin ( )f c khai trin Fourier

- Hermite :

( ) ( )0

n n

n

f Hf

=

=

( ) ( )n n E f H f = (2.21)

iu ny nht qun vi khai trin Fourier Hermite (2.21) v rng :

( ) 0nE H = , vi n > 0

Tnh l Parseval, ta c:

E[f2()]=

2

n=0

nf

(2.22)

Thng thng khai trin Fourier - Hermite l cng thc cho hm tt nh

trong khng gian u L2(R) vi o Lebesgue. Xc nh hm Hermite l:

( ) ( ) ( )1/ 2n n x x H x =

trong ( )x l phn phi Gauss (2.1)


57/77



Dng hm Hermite ( ){ }n l c s trc chun trong L2(R). Vi bt k

hm ( ) ( )2 g x L R , c khai trin:

( ) ( )n nn

g x g x= (2.23)

( ) ( )n nR

f g x x dx= (2.24)

Lin quan khai trin (2.21) n (2.24) , vi bt k hm ( ) ( )2 , f x L R , ta c

( ) ( ) ( )

1

2 F x f x x=

Khi khai trin (2.21) l tng ng

( ) ( )

( ) ( )

n n

n

n n

R

F x

F F x x dx

F

=

=

2.2.2 TNH CHT

Gi s ( ) ( )k f x B R , khi h s Fourier - Hermite ( ),n n f f H = l:

( ) ( ) ( )

( )

( )( )

1,

!

1,

... 1

n

n

kn

k

E f n k n

f

E f n k n n k

= > +

(2.25)

trong : E l k vng i vi o Gauss n v .

Chng minh:

Da vo cng thc (2.15) ta c

( ) ( ) ( )n n f f x H x x dx+

=


58/77



= 11

( ) [ ( ) ( )]nd

f x H x dxdx

n

+

= ( )11

'( ) ( )n x f x H x dxn

+

do php quy np ta d dng c cng thc (2.25)

Khai trin Fourier Hermite (2.21) c th m rng thnh a chiu. Vich s hu hn ( )1 2, ,..., d = vi thnh phn s nguyn khng m, xc nh

a thc Hermite nhiu bin bng tch v hng

( ) ( )1

d

i i

i

H x H x =

=

Khi ( ){ }H x l cs trc giao trong khng gian Hilbert ( )2 ,d dL R , trong

d l o Gauss d chiu trn dR .

K hiu ( )1 2, ,..., d = l vectngu nhin vi nhng thnh phn c lp.

Gi s l hm ca bin ngu nhin vi ( )2E q


59/77



CHNG 3

QU TRNH NGU NHIN DNG HERMITE

T khi nim v qu trnh ngu nhin Wiener kt hp vi cc a thc

Hermite ta s xy dng c qu trnh ngu nhin dng Hermite . Chng s tr

thnh c s trc giao ca khng gian cc qu trnh ngu nhin. V vy trongchng ny ta tp trung nghin cu v nu c mt sc tnh ca vi ngu

nhin i vi qu trnh ngu nhin dng Hermite.

V mt l thuyt chng c nhng tnh cht l th v cng c nhiu ng

dng quan trng. Ta thc Hermite mt bin chng 2 ta m rng a thc

Hermite hai bin chng 3.

Ta bt u khi nim v qu trnh ngu nhin dng Hermite

3.1 KHI NIM V QU TRNH NGU NHIN

DNG HERMITE

3.1.1 NH NGHA

a thc Hermite bc n l a thc xc nh bi :

( )( )

2 2n*

2 2

n

d, , 0,1,2....

! du

u un

t tn

t H u t e e n

n

= =

(3.1)

3.1.2 CC V D

Theo nh ngha trn, ta c :


60/77



Khi n = 0 : ( )*

0 , 1H u t =

Khi n = 1 : ( )*

1 ,H u t u=

Khi n = 2 : ( )2*

2 , 2 2

u tH u t =

Khi n = 3: ( )3*

3 , 6 2

u tuH u t =

Khi n = 4 : ( )4 2 2*

4 ,

24 4 8

u tu t H u t = +

..


61/77



3.2 TP TRC CHUN Y TRONG ( )2L R V ( )2 n

L R Trong phn ny ta xc nh hm Hermite chun suy rng.

3.2.1 NH NGHA

Vi m = 0, 1, 2, v [ ]0,t T , ta xc nh a thc Hermite suy rng

trong( )

( )

v a t

b t

bc m l

( ) ( ) ( )( )( )( ) ( )( )2 2m2 d, 1 exp exp -

2 ( ) 2 ( )

mm

m m

v a t v a t Q v t b t

b t dv b t

=

(3.2)

Cng thc (3.2) c gi l a thc Hermite suy rng vi hai bin (v, t)

Theo nh ngha t (3.2) ta s thu c

Khi m = 0 : ( )0 , 1Q v t =

Khi m = 1 : ( )( )

( )1 ,

v a tQ v t

b t

=

Khi m = 2: ( )( )

( )

2

2 , 1v a t

Q v tb t

= +

3.2.2 CC TNH CHT

Tnh cht 1:1' ( , ) ( , )

( )m m

mQ v t Q v t

b t= (3.3)

Ch :Vi m = 0, 1, v [ ]0,t T


62/77



o hm 2 v biu thc (3.2) theo bin v ta c :

( )1

21 ( )( ; ) ( ; ) ( ) ' ( ; )

( )m m m

v a tQ v t Q v t b t Q v t b t

+ =

(3.3)

Vi m = 1,2, v [ ]0,t T t (3.3) v ( 3.3)

Ta c

1 1

( )( , ) . ( , ) ( , ) 0

( )m m m

v a tQ v t Q v t mQ v t

b t+

+ =

Tnh cht 2:Vi bt k s nguyn m v k khng m ,

( )2

v - (t)exp ( ; ) ( ; )

2 ( )

0,

! 2 ( ),

k m

R

a I Q v t Q v t dv

b t

k m

k b t k m

=

=

=

(3.4)

Chng minh :

Gi s vi m k

Cho( )

2v- ( t )

( ; ) exp2 ( )

av t

b t

=

Khi :

( )( )( )

( ) ( ) ( )( ) ( )2

2v-a t

exp - , 1 ,2

kk k

kQ v t b t v t b t

=

S dng cng thc trn thay vo cng thc (3.4) ta c :

( ) ( ) ( )21 ( ) ( , ) ( , )k

k k

m

R

I b t v t Q v t dv


63/77



p dng cng thc tnh tch phn tng phn ta c :

( ) ( ) ( ) ( )

( ) ( )

1 1 '2

1 ( 1) '2

1 ( ) ( ; ) ( ; ) ( ; ) ( ; )

1 ( ) ( ; ) ( ; )

k

k k km m

R

kk k

m

R

I b t v t Q v t v t Q v t dv

b t v t Q v t dv

+

=

Tip tc cch ny ta c

( ) ( ) ( ) ( )21 ( ) ( ; ) ( , )k

k m k m m

m

R

I b t v t Q v t dv+ =

Ta xt 2 trng hp :

+ Trng hp 1:

Nu m < k th I = 0

+Trng hp 2:

Nu m = k , s dng cng thc (3.3) trn

( ) ( )( )

( )

22

2

2

2

v- (t)!1 ( ) exp - 2 ( )

( )

v- (t)! exp -

2b(t)

kk

k

R

R

ak I b t dub t

b t

ak du

=

=

= ( )! 2k b t

Vy tnh cht 2 chng minh xong.

3.2.3 NH NGHA

Vi m = 0,1.. v [ ]0,t T ta xc nh hm Hermite suy rng bc m l :

( )2

v- (t)( ; ) ( ; )exp -

4 ( )m ma

h v t Q v t b t

=


64/77



V ta xc nh hm Hermite chun suy rng bc m l

( )1

2( ; ) ! 2 ( ) ( ; )m mK v t m b t h v t

= (3.5)

3.2.4 TNH CHT

Tp hp hm }{0m m

K

=xc nh bi (3.16) l tp hp cstrc giao 2 ( )L R

Chng minh:

S dng cng thc (3. 5) trn vi mi s nguyn k v m khng m

( ) ( )( ) ( )

( ) ( )( )

1 1

2 2

21 1

2 2

,

( ; ) ( ; )

! 2 ( ) ; ! 2 ( ) ;

v - (t)! 2 ( ) ! 2 ( ) ( , ) ( , )exp -

2 ( )

k m

k m

k m

R

k m

K v t K v t dv

k b t h v t m b t h v t dv

ak b t m b t Q v t Q v t dv

b t

=

=

=

W


65/77



3.3 MT SC TNH CA VI PHN NGU NHINI VI QU TRNH NGU NHIN DNG HERMITE

3.3.1 NH NGHA

Cho{ }tW : 0t l qu trnh Wiener tiu chun mt chiu (chuyn ng

Brown ), khi ta xc inh cng thc Hermite bi cng thc truy hi sau :

( )( )

( )

( )

( )

1

*

0 t

*

t t

2*t

2 t

3*t t

3 t

4 2 2*t t

4 t

W , : 1

W , : W

WW , :

2 2

W WW , :

6 2

W t WW , :

24 4 8

H t

H t

tH t

tH t

tH t

=

=

=

=

= +

-----------------------------------------

( ) ( ) ( )* * *

1 2t t t t

1W , : W W , W , , 2,3...n n n H t H t t H t n

n

= =

Hoc ta c th xc nh cch khc bi cng thc truy hi sau :

( )( )

t

2 2*

t

W

x xW , : exp exp -

! 2t 2t

n n

nn

x

t dH t

n dx=

=

Vy ( ){ }*

tW , ,n H t n N l qu trnh ngu nhin v ta gi chng l qu trnh

ngu nhin dng Hermite.


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3.3.2 NH L

Cho ( )* *

tW ,n n H H t = l qu trnh ngu nhin dng Hermite. Khi vi m

nguyn v ln hn 1 ta s c vi phn ngu nhin

( )1 2 2* * * * *1

1

2

m m m

n n n n n

m md H m H d H H H dt

= +

(3.6)

chng minh nh l trn ta cn chng minh b sau

3.3.3 B :

i vi qu trnh ngu nhin dng Hermite ta s c

( ) ( )* *

1t t tW , W , Wn nd H t H t d = (3.7)

Chng minh:

Trc ht ta nhn xt :

( )

( )

2

2 n 2

n

0 0

2

u - tu dexp - exp exp -2t 2 2d

uexp -

2t

n

n

nn

n

t dutd

dt

du

= =

=

=

suy ra :

( )n 2 2 2

n

0

d u uexp exp exp -

2 2t 2td

nn

n

t du t

du

=

=

(***)

M theo (3.1)

( )( ) 2 2* u u

, exp exp -! 2t 2t

n n

nn

t dH u t

n du

=

=> ( ) ( )2 2 *u u

exp exp - ! ,2t 2t

nn

nn

dt n H u t

du

=


67/77



Thay vo (***) ta c :

( )2 *

0

exp u - ! u,2

n

nn

d tn H t

d

=

=

Vy theo khai trin Taylori vi hm2

exp2

tu

ti = 0 ta s c :

( )2 *

0exp ,2

nn

n

tu H u t

=

=

Mt khc , nu ta p dng cng thc It cho hm

( )2 *

t t0

exp W W , .2

nnt

n

tH t

=

= =

(3.8)

Ta s c t li l nghim ca phng trnh vi phn ngu nhin

( )t

s

0

W

0 1

1 W

t t

t

t s

d d

d

=

=

= +

T, ta c:

* * *

1s s

0 0 10 0

1 W 1 Wt t

n n nn n n

n n n

H H d H d

= = =

= + = + (3.9)

( ) ( )* *

1t s s

0

W , W , Wt

n n H t H s d =

Vy ( ) ( )* *

1t t tW , W , Wn nd H t H t d =

Vy b chng minh xong.


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* Chng minh nh l 3.3.2

p dng cng thc It cho hm ( )tX ,m

tY t X=

vi m nguyn , ln hn 1 v

( )*

t tX W ,nH t

T cng thc Ito

( ) ( ) ( )2

22

1( ) , ( ) , ( ) ( ) , ( ) . ( )

2

Y Y YdY t t X t dt t X t dX t t X t b t dt

t x

= + +

V (3.7) ta suy ra c (3.6)Vy nh l 3.3.2 chng minh xong.

V d :

Vi m = 2 t cng thc (3.6) ta c :

2 2* * * *

12n n n nd H H d H H dt

= +

(3.10)

Ch :Cng thc (3.8) cn c th thu c t nhn xt sau:

Gi s 1X v 2X c vi phn ngu nhin tng ng l:

1 1 1 t

2 2 2 t

W

dX W

dX a dt b d

a dt b d

= +

= +

Khi :

( )1 2 1 2 2 1 1 2.d X X X dX X dX b b dt = + +

Vi ( )*

1 2 tW ,n X X H t

S dng cng thc (3.7) ta c c cng thc (3.10)

3.3.4 H QU:

Cho ( )*

tW ,nH t l cc qu trnh ngu nhin dng Hermite, ta s c :


69/77



( ) ( )*

2t t

0

W , exp Wt

n

n

H t e

=

= (3.11)

Tht vy khi s dng cng thc (3.8) vi = 1 ta s c c (3.11)

3.3.5 CC TNH CHT CA QU TRNH DNG HERMITE

Cho ( )*

tW , , 1,2,3..., 0n H t n t = l cc qu trnh ngu nhin dng

Hermite , ta s c:

a) ( ){ }*

tW , 0n E H t = . (3.12)

b) ( ) ( )2 2* *

1t s

0

W , W ,!

t n

n nt

E H t E H s dsn

= =

. (3.13)

c) ( ) ( )* *

1s s

0

W , W , W 0n

t

n s E H s H s d

= . (3.14)

d) ( ) ( )* *

1 t t tW , W , Wn nd H t H t d + = . (3.15)

e)( ) ( ) ( ) ( ){ }

* * * *

s s s s 1 s 1 s00 0

W , W W , W W , W , 0,t t

t

n m n m E H s d H s d E H s H s ds

= =

, ,n m N n m (3.16)

Chng minh:

Chng minh tnh cht (a) v (b)Ta c :


70/77



( ) ( )*

2tW , 0, 1,2,3,...; 0n H t L t n t = >

v t (3.7) ta c :

( ) ( )* *

1t s S

0

W , W , Wt

n n H t H s d = (3.17)

Ta chng minh (a) v (b) i vi cc hm bc nhy, ( )*

1 sW ,nH s v gi s

rng ( )( )* *

1 1sW ,k

n n H s H = khi 1k k s s s + ;( )*

1

k

nH l F( )ks - o c v

F( )ks c lp vi - trng sinh bi chuyn ng Brown trong tng lai sau

thi im ks

a) Ly k vng 2 v (3.17)( ){ } ( )

( ) ( )( )

* *

1t s S

0

( )1 *

1 k +1 k

0

W , W , W

W s W s

t

n n

kn

n

k

E H t E H s d

E H

=

=

=

( ) ( )(( )1 *

1 k +1 k 0

W s W s 0kn

n

k

E H E

=

= =

do( )*

1

k

nH v ( ) ( )k +1 k W s W s c lp vi nhau

Vy ( ){ }*

tW , 0n E H t =

b) =>( ) ( )( ) ( ) ( )( ){ }

21*

( ) ( )1 S 1 1 k +1 k +1 j

, 00

W W s W s W s W st n

k jn n n j

k j

E H d E H H

=

=

Vi j < k


71/77



khi ( ) ( )k +1 k W s W s c lp vi( ) ( )

( ) ( )( )* *

1 1 j+1 jW s W sk j

n nH H

Ta c :

( ) ( )

( ) ( )( ) ( ) ( )( )

( ) ( )

( ) ( )( ) ( ) ( )( )

* *

1 1 +1 k j+1 j

* *

1 1 +1 j +1 k

W s W s W s W s

W s W s W s W s 0

k j

n n k

k j

n n j k

E H H

E H H E

=

=

(do ( ) ( ){ }k +1 k W s W s 0E = )Do :

( )

( ) ( )( )

( )

( ) ( )( )

( )

( )

221* * 2

1 1s +1 k 00

21 * 2

1 +1 k0

21 *

1 +1 k0

2*

1

0

W W s W s

W s W s

s s

t kn

n n k

k

kn

n k

k

kn

n k

k

t

n

E H d E H

E H E

E H

E H dt

=

=

=

=

=

=

=

Vy ( ) ( )2 2* *

1t s

0

W , W ,t

n n E H t E H s ds

=

Chng minh (c )T h thc (3.10) ta c :

( ) ( ) ( ) ( )2* * *

21 1t, s s s s

0 0

W 2 W , W , W W ,t t

n n nn H t H s H s d H s ds = +

Ly k vng 2 v


72/77



=> ( ) ( ) ( ) ( )2 2* * * *

1 1t, s s s s

0 0

W 2 W , W , W W ,t t

n n n n E H t E H s H s d E H s ds

= +

Theo tnh cht (b)

( ) ( )1

* *2 2

t s0W , W ,

n

t

n E H t E H s ds

=

nn

( ) ( ) ( ) ( )2 2* * * *

1 1 1s s s s s

0 0 0

W , 2 W , W , W W ,t t t

n n n n E H s ds E H s H s d E H s ds

= +

Vy ( ) ( ){ }* *

s 1 s0W , W , W 0

t

n n E H s H s d s =

Vy tnh cht (c) chng minh xong

Chng minh tnh cht (d) : ng thc (3.15) chnh l ng thc (3.7) m ta chng minh

Chng minh tnh cht (e) : da vo tnh ng c Ito ca tch phn ngunhin.


73/77


74

Lun vn thc s tonhc Trn ThVn Anh

KT LUN

Khai trin trc giao ca hm ngu nhin c nhiu vn hp dn, th v

cng nh nhng ng dng thc t ca chng trong cuc sng ca chng ta. Ti

rt mun nghin cu thm v a ng dng thc t ca chng vo lun vn ny

c c nhng kt qu tt p v mt l thuyt cng nhng dng vo thc

tin.Khi nghin cu v nhng vn v lun vn chng ta nu c mi

lin h gia a thc Hermite, qu trnh ngu nhin dng Hermite, a thc

Hermite suy rng bc m v mt sc tnh vi phn ngu nhin i vi qu trnh

ngu nhin dng Hermite.

Hng pht trin tip theo s nghin cu su v h s Fourier Hermite

suy rng v hm Fourier Hermite suy rng. Khi c c tp trc chun y

trong [ ]( )2 , 0,a b L C T . Ngoi ra cng c th nghin cu ti p php bin i

khng gian hm Fourier Wiener suy rng.


74/77



TI LIU THAM KHO

TING VIT

[1] A .D. VENTXEL

Gio Trnh L Thuyt Qu Trnh Ngu Nhin

NXB Mir Maxcova, 1987

( Bn dch t ting Nga sang ting Vit ca Nguyn Vit Ph v NguynDuy Tin)

[2] DNG TN M

Qu Trnh Ngu Nhin

Phn Mu

NXB i Hc Quc Gia Thnh Ph H Ch Minh, 2006

[3] DNG TN M

Qu Trnh Ngu Nhin

Phn I : Tch Phn v Phng Trnh Vi Phn Ngu Nhin

NXB i Hc Quc Gia Thnh Ph H Ch Minh, 2007

[4] INH VN GNG

L Thuyt Xc Sut V Thng K

NXB Gio Dc, 2000

[5] NGUYN BC VN

Xc Sut V X L S Liu Thng K

NXB Gio Dc, 2000


75/77



[6] NGUYN DUY TIN, NG HNG THNG

Cc M Hnh Xc Sut V ng Dng

Phn I : - Xch Markov V ng Dng

Phn II: - Qu Trnh Dng V ng Dng

Phn III: - Gii Tch Ngu Nhin

NXB i Hc Quc Gia H Ni, 2000 2001

[7] NGUYN DUY TIN, NGUYN VIT PHCSL Thuyt Xc Sut

NXB i Hc V Trung Hc Chuyn Nghip H Ni, 1983

[8] NGUYN DUY TIN, NGUYN VIT PH

L Thuyt Xc Sut

NXB Gio Dc H Ni, 2000

[9] NGUYN H QUNH

Chui Thi Gian : Phn Tch V Nhn Dng

NXB Khoa Hc V K Thut H Ni, 2004

[10] TRN HNG THAO

Tch Phn Ngu Nhin V Phng Trnh Vi Phn Ngu NhinNXB Khoa Hc K Thut H Ni, 2000


76/77



TING ANH[1] A. J. CHORIN

Hermite Expansion In Monte Carlo Simulations

J. Comput Phys, 8 : 472 482, 1971

[2] BERNT OKSENDAL

Stochastic Differential Equation An Introduction With Application

6th edition, Springer, 2005

[3] DEBNATH. L AND MIKUSINSKI

Introduction To Hilbert Spaces With Application

Academic Press ,1990

[4] F. H. MALTZ AND D . L . HITZL

Variance Reduction In Monte Carls Computations Using Multi

Dimensional Hermite Polynimals.

J. Comput Phys, 32 : 345 376, 1979

[5] R . H. CAMERON

The Orthogonal Development Of Non Linear Functionals In Series Of

Fourier Hermite Functionals.Ann Of Math, 48 (1947), 385 392,

[6] R . H. CAMERON

Some Examples Of Fourier Wiener Transforms Of Analytic Functionals.

Duke Math. J. 12 (1945), 485 488


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[7] R . H . CAMERON AND W. T. MARTIN

Fourier Wiener Transforms Of Analytic Functionals

Duke Math. J. 12 (1945), 489 507

[8] SEUNG JUN CHANG AND HYUN SOO CHUNG

Generalized Fourier Wiener Function Space Transforms

J. Korean Math. Soc. 46 (2009), No 2, 327 345

[9] WUAN LUO

Wiener Chaos Expansion And Numerical Solutions Of Stochastic Partial

Differential Equations

Californial Institute Of Technology Pasadena , California Defended May

2, 2006

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