On the mollifier approximation for solutions of stochastic differential equations

J. Math. Kyoto Univ. (JMKYAZ) 22-2 (1982) 243-254 On the mollifier approximation for solutions of stochastic differential equations By Shu Jia Guo (...
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J. Math. Kyoto Univ. (JMKYAZ)

22-2 (1982) 243-254

On the mollifier approximation for solutions of stochastic differential equations By Shu Jia Guo (Received Jan. 24, 1981)

§ O.

Introduction

Consider the following stochastic differential equation (SDE) on Rd -

o (X (t)).dW P(t)+bi(X (t))dt

dx= P=1

(0.1)

o-;,(X (0)•dW P(0-q—

0.

P=1

X(0) = x eR d

a 71 ( ; ' aip )(X (0)+bi(X (0)1dt,

1=1 , 2 ,..., d

with sufficiently smooth functions cil(x) and b ( x ) on R d . H e r e , ociW P(t) and •dWP(t) denote the stochastic differentials of the S tratonov ich type and of the Itô type respectively, a n d W (0= W (t, w )=(W P(t)), where W (t, w)= w(t), w e W6, is the canonical realization of the r-dim ensional W iener process on the r-dimensional Wiener space (W6, P s '): W 6' is the space of all continuous functions w: [0, co)--did such that w(0)=0 and P w is the r-dimensional Wiener measure on W . In tro d u cin g vector fields A 0 , A 1 ,..., A,. on R d by d

Ap ( x ) =

E

ai(x)

i= 1

fi

a X

d

A o (x )

E

1=1

b (x )

6

13=

4

r

a 6 .V

7

the equation (0.1) is also denoted by I d X (t)= ± A p (X (t))0dW o(t)+ A o (X (t))dt VI

(0.1)' X(0) = x.

If o (x ) a n d bi(x ) a r e C x with bounded derivatives o f a ll o rd e rs, the solution X (t, x, w) exists globally and for a.a.w(Pw), x , w ) is a diffeomorphism of R d for each t 0 (cf. [1], [3]).

244

Shu fia G uo

L et W (t)_ ( W (t)»» (5 >0) be an approximation of the W iener process W(t), i.e. th e process defined on (140 6, Pw) which consists o f sm ooth paths and which approximates W (t) a s ( 5 0 . Then w e can consider a dynamical system, i.e., an ordinary differential equation (ODE)

f )6(0 = ± A (X (t)) W (t )+ Ao (X (t)) 1

6

6

#=I

(0.2)

X6 (0) = x,

1

ddt

a n d we obtain a family (X ,(t, x, w)) of diffeomorphisms over R° defined by the solution of (0.2). It is reasonable to expect for a class of nice approximations that X 6 (t, x, w ) actually approxim ates X (I, x , w ) . I n f a c t, f o r the piecew ise linear approximation, this approximation of diffeomorphisms was obtained by Elworthy [2], Ikeda-W atanabe [3] and Bismut [ I ] , and for the mollifier approximation (a regularization by convolutions) it was discussed by M alliavin [4]. In particular, Malliavin called this approximation the transf er principle and regarded it a fundamental principle in studying the flow of diffeomorphisms X (t, x, w ). It seems difficult, however, to follow his proof in several points. M ain objective of the present paper is to give a rigorous proof of the mollifier approximation by modifying the method of [3] in the case of piecewise linear approximation. The author would like to express his hearty graditude to Professor S. Watanabe who kindly guided his research while he was staying at Kyoto University and gave him many advices in writing this article.

§ 1 . Mollifier approximation

L e t (In , Pw ) be the r-dimensional Wiener space and a r ----A(w6) be the usual a-field generated by the paths up to tim e t. Let p be a Cx-function with support in [0 , I] such that

p

O

and

p (t)d t = 1 . Upon choosing such a function, we set

for each S> 0 Wb(t)==

W i(t+ s,

w)p (A.-)

6 1

'

=1,

r

and call W (t)=(14/(0) a mollifier approximation of W(t, w). In order to emphasize the dependence of W6 on w, we often denote W6(t)= 14/6 (t, w). It is easy to verify the following properties of the mollifier approximation: (i)

t-> W6 (t) is Cx as a map : (0, oo)-+Ftd and

sup I 147 6 ( t, w ) - W (t, 1 0 1 -0 as 5 .1. 0 for every T > 0 and iv e t 0

0 St ST

-

K 24.

dYa(s)IIP]dt)

S hu f ia G ao .

252

Next we proceed to the case of ID:1=2. Set (t . • n 12.

02

x ' 11.)

x,

x

X

d•

2

Then 6 (s) W g(s)ds + cx:;'il

'fl( ,r,(s))1

(2.2 2 )

j 2

k=111=I JO

where (2.23)

=

14 eiras(s));‘,/ Y.5(s) ij, Ya(s)2 1 (s)d s

o k,I=I 11=1

with .

=

ah X) (

x•

a n d e ;(x)1, — l

1

02 OXkOXI

o, ( x ).•

If we denote a4.i.-12(1) as r

d

E

k,I=Ifi=1

[ i q ,5 ] - 1

( E

m=0

o- 73 (X 6 ((n2-1)(5))i,,Y 3 ((n2-1)(5) 1),

x Y 6 ((m -1)(5) i12 [ Wfla ((m+1)(5)— W l(m 6)])

(2.24)

O

+

[ t/ d )-1 ((m + i)6

r

k,I=1 11=1

(

Y ,(s)), Y A S P i

m=0 JmO

—a'i gX 6 ((M — 1 )(5 ))i I YA(M — 1)(5) 1), Y 0((M ,

E

Y ,(s)i),



1)(5) i1

2

2

] Wil ( S ) d S )

W g (S )d S

k,1=1 11=1[ ( / 0 1

H (t) + H 2 (1) -F 113 (t), 1

H 1 (i) can be estimated by the method used in the estimate of /,(t) as follows:

E [ su p IH (OP] :5- K25E

MY 0(4)0(s))114ds)-2—

0 5 t5 7 '

(2.25) K26

21

0

E [ s u p 11Y6( 1 )11 1 d S < K 2 7 < 0 0 . 0 5 tS T

As for H 2 (t), we estimate it by the method used in the estimate of J 2 (t) and obtain E [ su p 1112 (01P] .

0 5 t5 T

(2.26)

-

K2 8 E [ (

E

m=0

su p

05t5_T

K29 G CoC)

In a similar way, we can obtain

IIY0(1)11 ) ( C ,+ C + m - I C .,+ - 1 C m ) ) P ] 2

253

Stochastic differential equations

E [ su p IH (t)r]

(2.27)

K 3
0, we have

(2.33)

lim Pw ( s u p s u p MAT (t, x, w)— 6

.310

x, w ) > e ) = 0

O S tS T Ix IS N

fo r a ll T>0, N > 0 an d m ulti-index a. D EPA R TM EN T O F M A T H E M A T IC S , K Y O T O U N IV E R S IT Y . P R E S E N T A D D R E S S : D E P A R T M E N T O F M A T H E M A T IC S , F U D A N U N IV E R S IT Y , S H A N G H A I, C H IN A .

254

Situ fia G uo

References [1 ] J - M . B is m u t, Principes de mécanique aléatoire, to appear. [ 2 ] K. D. Elworthy, Stochastic dynamical systems and their flows, Stochastic Analysis (ed. by A. Friedman and M. Pinsky), 79-95, Academic Press, New York, 1978. [3 ] N. Ikeda and S. W atanabe, Stochastic differential equations a n d diffusion processes, Kodansha, 1981. [ 4 1 P. M alliavin, Stochastic calculus of variation and hypoelliptic operators, Proc. Int. Symp. SDE. Kyoto 1976 (ed. by K. It6), 195-263, Kinokuniya, 1978.

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