Malliavin Calculus for Processes with Jumps (Stochastic Monographs : Theory and Applications of Stochastic Processes, Vol 2)

STOCHASTICS MONOGRAPHS Theory and Applications of Stochastic Processes A series of books edited by Mark Davis, Imperial College. London, UK

Volume 1 Contiguity and the Statistical Invariance Principle P. E. Greenwood and A. N. Shiryayev Volume 2 Malliavin Calculus for Processes with Jumps K. Bichteler, J. B. Gravereaux and J. Jacod

Additional volumes in preparaJion

ISSN: 0275-5785 This book is part of a series. The publishers will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

MALLIAVIN CALCULUS FOR PROCESSES WITH JUMPS

KLAUS BICHTELER The University of Texas

at

Austin

JEAN-BERNARD GRAVEREAUX Universite de Rennes JEAN JACOD Universite Pierre et Marie Curie, Paris

GORDON AND BREACH SCIENCE PUBLISHER� New York London Paris Montreux Tokyo

Copyright © 1987 by OPA (Amsterdam) B.V. All rights reserved. Published under license by Gordon and Breach Science Publishers S.A. Gordon and Breach Science Publishers Post Office Box 786 Cooper Station New York, New York 10276 United States of America Post Office Box 197 London WCZE 9PX England 58, rue Lhomond 75005 Paris France Post Office Box 161 1820 Montreux 2 Switzerland 14-9 Okubo 3-chome Shinjuku-ku, Tokyo 160 Japan

Library of Congress Cataloging-in-Publication Data

Bichteler, Klaus. Malliavin calculus for processes with jumps. (Stochastic monographs; v. 2) Bibliography: p. Includes index. 1. Stochastic analysis. 2. Functional analysis. 1. Gravereaux, Jean-Bernard, 1945. II. Jacod, Jean. III. Title. IV. Series. QA274.2.B53 1987 519.2 86-31825 ISBN 2-88124-185-9 ISSN 0275-5785 No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publishers. Printed in Great Britain by Bell and Bain Ltd, Glasgow.

CONTENTS Introduction to the Series

vii

Preface

ix

CHAPTER I Results

1

Section 1

1

Introduction

The main results General setting and assumptions 2-b Existence of a density 2-c Regularity of the density 2-d Broadfunctions

Section 2

2-a

Section 3

6 6

10 13 16

One example

19

CHAPTER II Techniques

25

Section 4

4-a 4-b

4-c 4-d

Section S S-a S-b S-c

Toward existence and smoothness of the density for a random variable Integration-by-parts setting Iteration of the integration-by-parts formula Joint smoothness of the density More on joint smoothness Stability for stochastic differential equations Graded stochastic equations Differentiability Peano's approximation

26 27 29

34 36 44 45 50

54

CHAPTER III Bismut's Approach

59

Section 6

59 59

6-a 6-b

6-c 6-d

Calculus of variations The general setting The Girsanov transform Perturbation of the stochastic differential equation Explicit computation of DX

61

65 70

CONTENTS

VI

6-e 6-f

Section 7 7-a 7-b

7-c

Higher derivatives Integration-by-parts setting for

rc

Proof of the main theorems via Bismut's approach Introductory remarks Existence of the density Smoothness of the density

72 76

81 81 82 91

CHAPTER IV Malliavin's Approach

101

Malliavin operators Section 8 8-a Definition of Malliavin operators 8-b Extension of Malliavin operators 8-c Malliavin operators Oll a direct product

101

Malliavin operator on Wiener-Poisson space Malliavin operator on Poisson space 9-b Malliavin operator on Wiener space 9-c Malliavin operator on Wiener-Poisson space 9-d Malliavin operator and stochastic integrals

Section 9 9-a

Section 10 lO-a lO-b lO-c

Malliavin operator and stochastic differential equations The main result Explicit computation of U Application to existence and smoothness of (he density

101 104 108

112 112 116 117 118 130 130

140 143

147 147

Section 11 Proof of the main theorems via Malliavin's approach 11-a Introductory remarks 11-b Existence of the density 11-c Smoothness of (he density

149

Section 12

150

Concluding remarks

148

References

155

Index

159

Notation

161

Introduction to the series The journal Stochastics publishes research papers dealing with stochastic processes and their applications in the modelling, analysis and optimization of systems subject to random disturbances. Stochastic models are now widely used in engineering, the physical and life sciences, economics. operations research, and elsewhere. Moreover, these models are becoming increasingly sophisticated and often stretch the boundaries of the theory as it exists. A primary aim of Stochastics is to further the development of the field by promoting an awareness of the latest theoretical developments on the one hand and of all problems arising in applications on the other. In assodation with Stochastics. we are now publishing Stochastics Monographs, a series of independently produced volumes with the same aims and scope as the journal. Stochastics Monographs will provide timely and authoritative coverage of areas of current research in a more extended and expository form than is possible within the confines of a journal article. The series will include extended research reports. material derived from lecture courses on advanced topics, and multi-author works with a unified theme based on conference or workshop presentations. MARK DAVIS

vi;

PREFACE Since Malliavin introduced the new method in stochastic analysis, which now bears his name, much work has been done on various theoretical and applied aspects of the subject. However, essentially all this work has been concerned with analysis of continuous processes. It was thus very tempting to see whether significant results could be achieved for discontinuous processes using the same sort of analysis, especially after Bismut cleared part of the way in 1983. We started by extending Bismut's approach in a relatively short paper, then discovered that the original approach of Malliavin, Stroock and others was also feasible for discontinuous processes. The two approaches were compared and this work grew into the present monograph. This book provides several new results, but the emphasis is clearly on methods. JEAN JACOD

CHAPTER

I

RESULTS

Section 1: INTRODUCTION

r 19]

Malliavin

succeeded in proving some of Horman-

der's regularity results using purely probabilistic techniques. Recall the Problem 1: Under which conditions on the pair

(a, B)

(Pt(x,dY))t~O

of coefficients does the semi-group with generator

L

I

=

i

.

a

1

a2

i'

a~(x)--- + Z.I.B J(X)axidx. aX i ~,J J

have a density

Pt(x,y)

Pt(x,dy)

=

(1-1)

Pt(x,y)dy ? When

(0

is the density of class C in (t,x,y)

in

y

or even in

(x,y)

or

jointly?

The probabilistic argument is this. The measure Pt(x,dy)

is also the law of the solution

X~

of the

stochastic differential equation

~ = x where

bb T

+

J

t

a(Xx)ds + o s is the matrix

t

J

0 B

dimensional Wiener process (b

b(Xx)dW s s of 1-1 and T

0-2 ) W

is an m-

denotes the transpose of

b; we shall not worry here about the problems connected with the regularity of the representation B=bb T ). From the probabilistic angle the problem is therefore to discover under which conditions on the pair (a,b) of coefficients the distribution of the function

XX t

on the

Wiener space has a density or even a smooth density. Malliavin attacked this problem by transferring to

1

2

MALLIA VIN CALCULUS

Wiener space the analysis one would use to solve the corresponding problem on n d . Integration-by-parts plays a central role in this approach (integration-byparts formulae on Wiener spaces were actually known for some time:

see Kuo [15]

and Haussmann [ 11]). Malliavin

calculus was developed and extended by several authors. Stroock [24], [25], [26]

establishes the central inte-

gration-by-parts formula for the number operator, which is the generator of a diffusion semi-group on Wiener space (infinite-dimensional Ornstein-Uhlenbeck process). Similar approaches were given by Shigekawa [23]

and

Ikeda and Watanabe [ 12]. Stroock has also several regularity results not accessible to analytic treatment. Bismut [6]

uses Girsanov's theorem and flows -

is, deterministic semi-groups -

that

on Wiener space to

obtain a "directional" integration-by-parts formula. His method was simplified to some extent by Fonken [3], [4], [9]

and Norris [21].

It has the advantage, reco-

gnized and exploited by Bismut himself [7], of generalizing rather easily to integro-differential operators of the form

L1

=

Kf(x)

L+ K

= f[

f (x+ y) - f (x)

1.

Here

0-3)

-? a! . (x) Y i) K (x, d y) . 1.

d

that integrates is a positive kernel on lR 2 (a "Levy kernel"). Again a memthe func t ion y ~IYI ber

K

P~(x,dy)

of the semi-group with generator

L'

is

x

the law of the solution

X t of a stochastic different-

ial equation, namely t

t

t

J x )ds b(Xx)dW c(Xx ,z)dP x+a(X 0-4 ) o s s 0 E so s driven by time, Wiener process Wand the "compensated Xxt

=

Poisson measure" ii

+J

+J J

of a Poisson measure \l on

lR+xE:

INTRODUCTION

3

E is an auxiliary space, with a positive a-finite measure G,

the intensity measure of

~

is v(dt,dx)=dtxG(dx).

1- 3 and 1-4 are connected through

Moreover,

B = bb T

,

K(x,A)

=

f

(1-5 )

lA,{O}(c(x,z»G(dz). E

Bismu t

can thus address:

(7)

Problem 2:

Under which conditions on the triple (a,b,K)

of 1-3 does P~(x,dy) density,

admit a density p~(x,y),

a regular

a joint regular density?

Bismut solves this problem in a very special case

L'

where the Markov process with generator tribution which, truction)

for any starting point,

has a dis-

is

(by cons-

absolutely continuous with respect

to the

distribution of a fixed process with stationary increments whose semi-group admits densities. of the existence of a density for not arise.

technique another,

L'

does

therefore

The regularity of these densities,

is a difficult problem.

when

(P~)

The question

He [8]

however,

has also solved with his

closely related,

problem,

namely

is the generator of a continuous diffusion with

boundary;

although the process is continuous,

a Levy

kernel arises in connection with the excursion process (see also Leandre [18]). Here we investigate a Problem 2,

problem closely related to

in a much more general context than Bismut;

there is however a notable difference (and, as

the non-

probabilistically minded analyst might say, unfortunate) : Problem 3: of 1-4 does

Under which conditions on the triple P~(x,dy)

admit a density,

a jOint regular density?

a regular

(a,b,c) densit~

4

MALLIA YIN CALCULUS

The main results are described in Section 2. We give a reasonably (?) general condition on (a,b,c) for the existence of a density: basically, diffuse part (expressed by ressed by

it says that the

b) and the jump part (exp-

c, or rather its derivatives oc/az.) must J.

"fill" the whole tangent space at every point

xElR

d

thus it essentially is a condition of non-degeneracy. When XX is continuous (i.e. c=O)

this amounts to non-

degeneracy, or strict ellipticity, of the matrix B= T bb : we are far from recovering the full force of Hormander's Theorem (it would be possible however to get "weak Hormander" conditions: see the works [17], [18] of Leandre, who obtains such results in a particular case). Then we state a uniform non-degeneracy condition which yields a given order of differentiability of the density. In Section 3 we sketchily develop an example, closely related to [17] . Now,

the emphasis of this monograph is not really

put on the above-mentionned results, but on the methods: we apply and extend Bismut's method; we also extend Mal1iavin-Stroock's method,

introducing to that effect

a "Malliavin calculus" on Poisson space. It is worth mentioning that the two methods give essentially the same results

(with a slight bonus for Bismut's one), as

far as Problems 1 or 3 are concerned. Chapter II is concerned with some useful techniques and more precisely two of these: 1) A general "integration-by-parts" setting is introduced in Section 4, and put to work for obtaining (smooth) densities of random variables. This is purely abstract, without reference to neither Poisson measures

INTRODUCTION

5

nor Wiener processes (we advise to read §§3-a,b only). 2) Some rather general

stabil~ty

and differentiabi-

lity properties for stochastic differential equations, in Section S. The results are by no means novel, but they are adapted to our needs. Chapter III is devoted to Bismut's method

(we have

already presented a version of the I-dimensional case in [5);

see also [1], of course!):

the "calculus of va-

riations" is expounded in Section 6, while Section 7 contains the proofs of the main theorems. Finally, Malliavin-Stroock's approach 1s presented in Chapter IV (some of the results have been announced in [10]). Malliavin's operators are presented in Section 8, in a rather abstract manner (not related to differential equations), and following Stroock [24), [25]

rather closely. Sections 9 and 10 give applica-

tions to Wiener-Poisson space and to stochastic differential equations. Section 11 provides another proof of the main theorems, and Section 12 is devoted to comparing the two methods and to further comments.

Section 2:

§2-a.

THE MAIN RESULTS

GENERAL SETTING AND ASSUMPTIONS

The time interval is the bounded interval [O,T]. Despite first appearances,

it will make our non-degeneracy

condition easier to state and to interpret if we consider 1-4 with several Poisson measures driving, of only one -

instead

just as an m-dimensional Wiener process

is but a collection of

m

independent I-dimensional

ones. Accordingly, we consider the 2-1

HYPOTHESES:

(fl'£;'{£t)tE[O,T]'P) is a filtered space

endoUJed with: - a standard m-dimensional Wiener -process w=(W i

)

. i <m'

a Poisson random measure ~a= on [O,T]XE , where E is an open subset of a a a ~Ba with infinite Lebesgue measure. The compensator - for

~

l~a~A>

(w;dt,dz)

va of ~a is of the form dtxGa(dz) > where Ga denotes Lebesgue measure on Ea . The ncompensated Poisson measure n Do is given by

- another Poisson measure ~=~(w;dt,dx) on [O,T]XE, with intensity measure v(dt,dz)=dtxG(dz); here G is a positive o-finite measure on a measurable space (E,&) and D=~-v is the compensated measure; - the random elements

i

(W '~a'~) ar~ independent.

Next, we are given a family of coefficients:

6

THE MAIN RESULTS b = (B ij ) c ex =

- ; d x E ..... ]R d

(c!) 1

C=(Ci)l
For each x

f o

+

lRd -+ lRdSlR m

1
~i~d

7

lRdxE-+]Rd. we consider the following equation: t

t

a(Xx

s-

J b(Xxs- )dW s 0

)ds +

t

LJ f

+

c

ex 0 E t

f

+

o

ex

( XX

s-

, z ) ii ( d s , d z ) ex

(2-2)

!c(Xx ,z)O(ds,dz). E s-

The assumptions made below will ensure the existence

XX

and uniqueness of a solution

to 2-2 as an

lRd-va-

lued cad lag process (strong solution). In fact,

..;<

is

a strong Markov process with generator:

L'

=

L

+ IK a a

Lf (x) =

1. a i °

+

K

with

a 1 i a2 fa (x) (x )-"- f (x) + -2 B J (x) " "xi i,j "xi Xj

L

K f (x) = / {f (x+y) - f ex

O

L yo i

(x) -

1.

Kf(x) = /{f(x+y) -f(x) -

hi i

The connection between

th~

-f-f (x)}K (x, dy) xi ex

~f(X)}K(X,dY). xi

Ka , K on the one hand, and the coefficients of 2-2 on the other,

matrix

(2-3)

B

and kernels

is given by

B(x) = b(x)b(x)T

~

Ka(x,A) = K(x,A) =

!

lA_{O}(ca(x,z»Ga(dz)

(2 -4)

lA_{O}(c(x,z»G(dz).

E

2-5

REMARK:

It should be noted that 2- 3 is not any

8

MALUA VIN CALCULUS

more general than 1-3;

it is always possible to aggre~~

gate the Poisson measures

z)

son measure on

~

and

~

into a single Pois-

By the same token,

+ E.

it is no

restriction to a~sume that G(l is Lebesgue measure; any ~

ded G

CL

2-6

~d can be written in the form 2-4, provi-

on

kernel K

has infinite mass.

REMARK:

In 2-2, dW t means Ito-differential. This

is contrary to the customary use of Stratonovitch differentials in the present context,

but it is made nece-

ssary because the process XX is not continuous. In many formulae, arrays appear;

vectors, matrices or higher order

in general, we shall not show summation

indices or symbols, unless there is ambiguity (for example in 2-2,

I l"::J..::m . a

r

the second integral really is

ftb·j(X X )dW j ). 0 ss

If

a

is a vector or a matrix,

denotes its transpose. for any finite array

lal 2

a,

is the sum of the squares of its entries; and

det(a)

denotes the determinant of the square matrix a.

We d enote b y

the space of all functions on ween

o

bounded.

r (R n ) Cp

r

( resp.

C rb ( Rn) ,resp.

Cor ( Rn))

times continuously differentiable

~n, whose derivatives of all order bet-

and

are with polynomial growth Crespo resp. with compact support). If fEC;(R n ), Dxf

stands for

vector C
xm

the

r

l'OlJ

we denote·by Dxf and Dzf the row vectors

((If/axi)l
THE MAIN RESULTS

of times.

9

To do this, we clearly need some reguiarity

on the coefficients of Equation 2-2. Here are the

assu~

ptions (below, rElN*). 2-7 ASSUMPTION (A-r).

(i) a and bare r-times differentiable ~ith bounded derivatives of alZ order bet~een 1 and r. d (ii) c Cl is r-times differentiable on lR xE , and c

a

(0,.)

sup sup

x

n2

E

~p~

L p (E

a

Ion c (x,.)1 E r,2 xn

Cl

,G )

a

~P

I D n + q c (x, z) I < xnzq a

x,z

(iii) c is

<'" co

for

LP(E,G) a

a

l
for 1~n + q~r and q~ 1 •

mdxE-measurable; c(.,z)

is r-times

differentiable, and c(O,.) E r,2 sup

x

~P
IDxn n c (x •. ) I

LP(E,G)

E

Ii 2

~p


L P ( E , G)

for 1
We shall always assume at least (A-2). which will garantee that 2-2 has one and only one solution (unique up to evanescent sets): see e.g. Section 5 below, or [ 13] . We end this subsection fixing notation.

~

denotes

the predictable o-field on I1x[ O,T]. We define the "maximal process"

H*

of a process

H

by

(2-8) For stochastic integrals we use the standard notation of [13]

or [20], for random measures those of [13]. The

stochastic (indefinite) vector

i

H=(H )1
H*W =

L Hi~Wi i

similarly, H*t

integral of a predictable row

with respect to

LJ

HidW i . s s denotes the process =

i

0

W

is written:

10

MALLIA YIN CALCULUS

h t (w) =

f

H (w)ds • s

o

For a measurable function

V

on nx[O,T]xE we set

t

f f

V~~ (w) = V(w,s,z)~(w;ds,dz) tOE when this integral exists, and similarly for

V

V*v a • When

is

~x~-measurable, V~P

stochastic integral process

ii =

~- v

i f and

x

=

+ a(

Jfo

+

+ b(

t

V

lP. \i T

With this notation,

~)

V~~a'

we can define the

of

(I V I 2 1\ I V I)

n 1 y if

V~Pa'

larly for

xx

0

V~v,

relative to

<'"

a • s ., and s i mi -

2-2 reads

X~ ) II" W

L c a (Xx- )1til a

(2-9) + c(Xx_hoP

a stands for c a (~_ (w) ,z). We usually prefer a 2-9 to 2-2 as it is somewhat more compact.

where c (~)

§2-b. EXISTENCE OF A DENSITY In the sequel, a discriminating role is played by the following functions on

~d

(resp.

~dxE). They take

values in the set of symmetric nonnegative dxd matrices: B(x}

(2-10)

b(x)b(x}T,

{( I + Dc) - ID cDc T ( 1+ Dc) - 1, T } (x, z)

C (x,z) a

=

x a

{

z a z
i f I+D c

o

x a otherwise.

Here, I is the dxd identity matrix. We may set N

x

N
xz

= kernel of B(x) in

=

~d (2-11)

kernel of Ca(x,z)

in

equivalently, N

x

o~thocomplement of the image of ~

under the dXm matrix b(x),

~m in

1 I

THE MAIN RESULTS

11

Ba

orthocomplement of the image of ~ )' in ~d under the dxB matrix (2-12) -1 a a d {I + 0 c (x, z) } 0 c (x, z). and N = lR x a z a xz if 1+0 c (x,z) is not invertible. x a The non-degeneracy assumption that will garantee the existence of a density is this:

Po r each a= 1 •••.• A there is a BoreZ subset r of lR xE such that, if r a,x is the a a x-section of r a in Ea and if ~A..::cS..::cS-=U.:.:M:..:.P..:.T..:.I-=O:..:.N,----,(-=B:...L):

2-13

d

Wa _ { -

x

u zE [ lR

d

a,x

then

N;z

{o}

G

if

G (r

a

a

an

for

(r

if

a,x

a,x

)=+cc ) <+"',

xElR d •

The first of our main theorems says: 2-14

THEOREM: Under (A-3) and (B), the variabZes

have a density

for aZl

y-Pt(x,y)

Note that the orthocomplement of N d

the vector subspace of

xElR x

d

x~

,tE(a,T].

(resp. Na ) is :xz

spanned by the eigenvectors

~

of B(x)

(resp. C (x,z» corresponding to positive eigena values. Thus an equivalent, and perhaps more appealing way to state Assumption (B) is this: 2-15

ASSUMPTION (B'):

(~(B»

is a Borel subset ra of Perty: let I(x)={a:G (r a

~

a

d x

and for each choice of z 'in r spanned by d equals ~ .

1.

(N)

x

a

Por each

'

a,x

and the family

In other words,

a=l, ... ,A there

xEa with the following prod )=+oo}. then for each xElR

for each xElR

,the vector space

{(N a

x. Za

d

I )-} EI ( )

a

x

and each choice z

a

12

in

MALLIA VIN CALCULUS

r a,x

when aEI(x),

the symmetric nonnegative matrix

B(x) + IaE1(x) Ca(x,za) tion G (r

is nondegenerate.

The assump-

)=+00 implies that in every finite interval

a,x the Poisson measure Ua has infinitely many jumps whose size belongs to r . Phrased thus, it becomes a,x clear that the true nature of Assumption (B) is a conCI.

(s,t)

dition of non-degeneracy on the whole collection (B, C ).

a

2-16

REMARK: This theorem, as well as the three other

main theorems below, are proved in Section 7 via Bismut's approach. As a matter of fact,

one can and will

prove them under assumptions called (A'-r), which are slightly less stringent (and much more complicated to state) than (A-r):

see 6-3.

Another proof of these theorems, via MalliavinStroock's approach, 2-17 REMARK:

is provided in Section 11.

Because of

(A-r-(ii»,

{I + D c

( x , z) }

-1

x a I , outside a set of G -

is as close as one wants to fin it e measure. Hence (N a )1... is "very clos e" to thea B xz image of m a under the linear map Dzca(x,z), this being the space in which the size of XX at

t

c (x' ,z) of a jump a is most likely to sit, given that XX =x' t-

and that Ua has a point at (t,z). Thus it would have been somehow more intuitive if in condition (B), (I+Dxca)

- 1

DzC a could be replaced by

Dzc a alone. Unwe could not achieve this substitution, due mainly to the fact that the set Wa in 2-13 is not x closed, and in spite of the smallness of Dxca. fortunately,

2-18

REMARK:

As seen above,

(B) allows· for a genuine

interplay between Wand the u 's. However, if c =0 for a a all a then (B) reduces to nondegeneracy of B(x) for all

THE MAIN RESULTS

x:

13

this is rather crude, and far from "weak Hoimander

conditions". A (relatively) simple addendum to the proof in Section 7 would indeed provide for conditions closer to Hormander's conditions, as Leandre did in in a particular case (namely,

[171

the case expounded

in Section 3 below)o 2-19

REMARK:

(B)

also shows the point in separating

one driving Poisson measure smoothness conditions on the c

Actually a ' s.

a

the

being much weaker than on

is often possible to chop an in-

it

finitesimal generator L ' c

c

from the others,

11

into pieces as in 2-3 wi th the

's that are very regular and a residual

c

that is

less regular. At this juncture,

it should be noted that Bismut

in the particular case he studies,

[7],

is asking for

only two degrees of differentiability in

z

for the

c

's, whatever r is in (A-r). This is an important a improvement upon (A-r), which comes from a difference

in the "iteration procedure" which is basic to the method. §2-c. REGULARITY OF THE DENSITY We begin by introducing some functions that might look strange at first sight; we shall make a stab at explaining them in the next subsection. 2-20

DEFINITION: A measurable function

is called numbers,

(~,e)-broad,

where

if the following

Ya(~,e,fa)

-- J'" ds o

This means that

s 1;-1

fa

~

and S

ar~

~ [0,00) a two positive

f

a

E

integral is finite: -sf (z) exp {-Sf (l-e a )dz} Ea does not go too fast to

(2-21)

0

at

14

MALLIA VIN CALCULUS

infinity. G (f

a

E

a

For example, we shall see soon that it implies

= +"".

>0)

(2-22)

For each a, we also consider a function P : a [0,"") with the following properties:

0.

2-23

Properties on Po: P a po(z)

+

0

as

z

is of class

c:,

and

goes to the boundary of Eo' and

I Dr p I EL 1 (E ,G) zr a a C/.

for a 11

rElN.

(Since Eo is a countable union of 8a-dimensional rectangles,

it is always possible to find functions

meeting 2-23 and being strictly positive). (SB-(~,8»: There exist two constant and for all C1=l, ••• ,A a (~,8)-broad function and a function P meeting 2-23, such that for alZ

2-24 pO, f

a

ASSUMPTION

:S~O,

d

x, yElR

Q

,

(2-25)

This is evidently a uniform non-degeneracy condition ( local uniform(!) would suit better). It clearly implies (B): simply take r = ~dx {f a >O} and use 2-22. a

ASSUMPTION (SC): There is a oonstant

2-26

that Idet {I+D x c a (x,z)} I ->~ identicaZZy.

and

~>O

suoh

Idet {I+Dxc(x,z)} I_>~

THEOREM: Let rElN* and tE(O,T). A;sume (se) and

2-27

either (i)

(A-(r+d+3»

and (SB-(Z;,e»

with 8
and

THE MAIN RESULTS 1;'

2d(r+d+1) [tie]

( [ ~] e

e'

d enotes t h ipart t of ) e nteger

(ii) (A-(r+3» 2d 2 (r+1) 1;> [tie] • Then,

the density

15

and (SB-«(;,e»

YM.-')

with e
of the

pt(x,y)

P.V.

XX e=ists t

and is of class cr. 2-28

Let rE1N* and

THEOREM:

tE(O,T].

Assume (SC) and

either (i) (A-(r+2d+3» 2d(r+2d+l) 1;> [tie] , (ii) (A-(r+3» 4d 2 (r+1) 1;> [tie]

and (sB-(t;,e»

and (SB-(I;,e»

Then (x,y)JV->Pt(x,y) Le t

2-29 THEOREM:

rE1N* and

4d(r+2d+2) [tie] ; assume

1;>

than

(SC»: !det{I+uD c X

ex

with 8
is of cZass cr. tE (0, T] .

(i) Assume (A-(2r+4d+6» and

with e
and (SB-(I;,6»

with 9
also the following

(stronger

(x,z)}!>t;',!det{I+uD c(x,z)}!>t;' x -

(2-30)

for all uE[O,l], where t;' is a constant. Then

(s,x,y)~ps(x,y)

is of class C r on [t,T]xlRdx]Rd.

(ii) Assume (A-(2r+4», and ! 0) YTB(x)y ~ ~!y! 2 l(l+!x e:>0 ..

O~O.

cex~O,

c50, and

identically for some constants

Then (s,x,y)lv--> ps(x,y) is of class

d

c r on

d

[t,T]xlR x]R. A (I;,e)-broad function is and

8'~8:

ons

(i) and

:t;',e')-broad for

I;'~t;

this is consistent with the form of conditi(ii)

in 2-27,2-28 and 2-29.

condition in 2-27-(i),

say,

The broadness

becomes more stringent as

16 T

MALLIA VIN CALCULUS

increases and as

t

decreases. In fact,

read the theorem as follows: holds for some

~,e;

t i a b 1 e for a 11 in t e g e r s

r

t~e, is wi t h

times differe~

r

r <[

e] t

~

/2 d - d - 1. 0 r

the density of X~ is

rEJN*,

ferentiable whenever [t/e]>2d(r+d+l)/~

(SB-(~,9»

suppose that

under suitable regularity conditi-

ons the density of x~, with also: given

one could

t

and

r

times dif-

is large enough to satisfy t~e.

The phenomenon of regularity increasing with

t

has already been encountered by Bismut [7], As a matter of fact,

the same phenomenon also occurs for the exis-

tence of a density (although our method is too crude to show this):

one can construct a process

stationary independent increments,

X

starting at

with

0, such

that Xl has a singular distribution and X2 has a density ([22],[28). Note that the assumptions theorems are not,

(i) and

(ii)

in these

in general, comparable, because the

broadness condition in (ii) is stronger than the one in (i)

(except for

d=l, or

d=2

and

r=O).

§2-d. BROAD FUNCTIONS ~e

devote this subsection to some comments and some

examples of broad functions.

Firstly,

the following

statements are obvious: 2-31

Any function bigger than a is

(~,e)-broad

(~,e)-broad

function

itself (if it is measurable, of

course) . fa

is

(~,e)-broad,

then so is

yf

for y>O.

2 - 3'2

If

2-33

Any strictly positive constant function on Ea is

a

THE MAIN RESULTS

17

for all z;;>0,9>0 (recall that· G (E ) a a =+00 by hypothesis. (z;;,6)-broad,

2-34 LEMMA: If Ga(C)<~, then fa is (z;;,9)-broad if and only if f 1 is (z;;,6)-broad (this obviously implies a Cc

2-22). Proof. We have -6G

e

a

- G ( C) <-

a

o

s

(1 - e

(z) a

) dz ,

hence

(C) Y (Z;;,6,f

a

I

-sf

r

_. C

Z;;-1

ds

1

a Cc

)

ex p {- eG ( c ) a

-sf

(z)

61

(l-e

)

is obvious.

a

)dz

< Y (Z;;,6,£ ),

a

wh i ley

a

a

(Z;;, 6 , f

)
a -

a

(Z;;, 6 , f

1

a Cc

2-35 Example of broad functions on Ea=(n,~)

or Ea=(-~,n)

~:

for some nE~.

Let 8 =1 and

a

Then one easily

checks that f

a

(z)

=

IzlY e- 01zl

In particular,

IzlY

a

o<~ Z;;

instead of

(Z;;,6)-broad iff

O<~,YE~.

is (Z;;,6)-broad for all Z;;>O,

for any real y. When E with

is

=~,

6>0,

we have the same result,

o<~1; •

2-36 Example of broad functions in higher dimension: Ba~2, and suppose that the boundary of Ea consists

Let

in a finite number of hyperplanes. Then f

IzlY e- 01zl ( i)

is

(1;,6)-broad in exactly

o~ and E Z;;

a

(z)=

th~ee

cases:

is contained in a cylinder with

(8 -I)-dimensional bounded basis, and extends a

to (ii)

infinity on one side of the basis;

o~ 1;

and E is as above, but extends to infia nity on both sides of the basis;

18

MALLIA YIN CALCULUS

(iii) oElR, yElR, and Ea is contained in no cylinder with bounded (8 -I)-dimensional basis. a

(Observe that 2-35 is but a special case of (i) or (ii»

.

COROLLARY: Suppose that each E is limited by a 8 (l finite number of hyperplanes in lR a In order that (SB-(~,e» be met, it suffices that there exist E>O, o~O, and for each a=I, ... ,A a reaZ number 0 such that a 0a<9/~ (resp. 0a<28/Z;, resp. 0aElR) if Ea is like in 2-36-(i) (resp. (ii), resp. (iii)), with 2-~

T y

B(x)y +

T

I

~

Proof. Let

Iy I

~

1 a

+

Pa(z)=lzl a if f (z)=e-oalzlp (z) a a from 2-38.

[0,=)

~

1+ x

E~=Ean{ZElRSa: Izl~l}.

a function p :E _S

0

inf zEE (y Ca(x,z)y)e a a a 2

Iz I (2 -3 8 )

10

One can easily find

meeting 2-23, such that

zEE'. Then, due to 2-36 and 2-34, a is (~,8)-broad, and 2-25 follows

Section 3: ONE EXAMPLE

We give here an example of how the main theorems can be put to work on the original Problem 2, when the operator L'

is given by 2-3, with the kernels K

a

sit-

ting on smooth curves. This example is essentially the case examined by Leandre in [ 18], but the results presented here are slightly different, and essentially weaker. in

More precisely, K (x,.) admits a smooth curve r (x) d (l a m as its support, on which it is absolutely conti-

nuous with respect to the arc-length measure. We suppose that the curves [(lex) are nicely parametrized by [ 0, 1] :

3-1

There is a C

1

mapping

d

Ya:m x( 0, 1]

that [a(x) is the image of (0,1] We suppose that for

Ya(x,O)=O

~ lR

d

t""">v , a (x,t).

under

and that

such

y

ex

(x,t)~O

t>Ya(x):=inf(s: Ya(x,s)~O), and that

Dty (l (x,y a (x)HO. Hence the curve leaves the origin at time

t=y (x)

a

and

never returns

(as the reader will immediately notice,

these are no

restrictions for the problem at hand);

also that Dty (x,y (x))~O can always be achieved by a a 0. suitable parametrization. Assume also that (3 -2)

K (x,C)

a

where:

19

20

MALLIA YIN CALCULUS

3- 3

go is of class C for all

and

x

1

d

on

lR x(0,11

go(x,t)~~

and

for all

Jo 1 g a (x,t)dt='" x,t

and some

constant 1;>00 In 3-3 the assumption

is a most unfortunate

ga~1;

restriction (see [51

for a discussion of this point in 1 J g (x,t)dt=oo Dad a restriction, and does not mean tfiat Ka(x,~ )='"

a similar context). The assumption that is

~

but merely implies the equivalence ( 3-4)

Now to apply our theorems we must obtain functions c

a

satisfying 2-4 with suitable E • A simple way goes 0

as follows.

Set 1

g (x,z) = a

J

g (x,s)ds ,

This is a C 1 function on

8a

(x,.)

ZE[

a

z

lR d x (0, 1]

is a bijection of (0,1]

by ga'(x,.)

its inverse:

0, 11 •

( 3- 5)

d b ecause an,

t: 0...

3 -3,

onto [0,""). We denote

g'(x,g {x,z)=z a a

for

zE(O,l].

Then set E

a

c

(0 , "') ,

a

(x,z) = y (x,g'(x,z». a

a

(3-6)

A simple computation shows that 2-4 holds, and that DtY -(----a)(x,g,(x,z» ga a D c

x a

(x,z)

(3-7)

D y

x a (x,g'(x,z») a

D y

1

+

J

Dx g (x, s) d s g'(x,z) a

(7) (x,g~(x,z». a

a

Finally we are also given

a,b,c

will use the assumptions (A-r)

and

as in 2-4. We

(SC) of §§2-b,c, the

corresponding conditions on (y ,g ) being easily (altha

ough tediously!)

(l

obtained via 3-7. Observe that the

last condition in 3- 3 plays a central role in obtaining

LEVY KERNELS SUPPORTED BY SMOOTH CURVES (A-r)-(ii)

for c CL (see [5]

21

for a more detailed discus-

sion of these assumptions.)

3-8

THEOREM: Assume (A-3)

with the Ka 's as above. Assu-

me al.so that foY' each xElR d

the vector space spanned by

a'll eigenveators of B(x) = b(x)b(x)T corresponding to positive eigenval.ue8~ together with all the tangent veators to the curves r (x) at the origin for those a d a d

for whiah K (X,lR )=+"', is equal. to

JR • Then

a d a density Pt(x,.) for al.Z t>O, xElR • Proof.

x

Xt

admits

By 2-14

it suffices to prove that (B'j (see 2-15) holds. With the notation 2-12, Na is the orthoxz complement of the vector U (x,z)= -1

a

(I+D c (x,z» D c (x,z) x a z a and U =0 otherwise.

if

CL

Let

x

I+D c x a

is

invertible,

d

be such that K (x,lR ) ="'. The function 2 CL (x,z) is in L «O,=),dz) and it has bounded

k(z)= D c x CL derivatives,

hence

lim z t

CD

k(z)=O. Moreover 3-7 yields

(D z c Cl liD z c a I)(x,z) = -(Dty a IIDtY a I)(x,g'(x,z» if a DtY (x,g'(x,z» -f0; we also have lim t g'(x,z)=O and a a Z"'CL y (x)=O (see 3-4) and Dty (x,O)-fO. Using the definition a a of Va and the property k(z)+O as zt"', we obtain that La(x,z)fO for U

a

lit I has: a

z

big enough and that the unit vector

U

lim zt = -ru:-r(x,z) a

outward unit tangent vector (3 -9)

to Now,

r CL (x)

at

O.

our assumptions and 3-9 imply that there are d measurable functions E :lR + (0,"') such that for any a choice of z E(E (x),"') the vector space spanned by a a d (N ).l.and the U (x,z), with aE1(x):={a:K (x,lR )==} x da a CL is equal to lR • Therefore (B') is met with r a

22

MALliA YIN CALCULUS

{(X,Z)ElRdXE

a.

:a.EI(x) and Z>E (x)}. a .

Next, we give a smoothness result, aiming to a sufficient condition that pertains essentially to the geometrical properties of the curves r

Il

(x). The following descriptio~

uniform non-degeneracy condition is of this 3-10

ASSUMPTION

x,yElR

3-11

d

There is E>O such that for all

(SB'):

,

P(~) such that:

IDtYIl(x,g~(x,z»

Y (x)=o and z>p(n) a. Dtya.(x,O) as ztoo

there is a constant

- DtYa(x,O) I~~ whenever

(note that

Dty (x,g'(x,z» + Il a when Ya(x)=O; so this is ,an assump-

tion on the uniformity in 3-12

~>o

ASSUMPTION (D): For each

x

for

THEOREM: Assume (A-(r+3)L

this convergence).

(SE').

(D).

(SC). As-

sume also that z>r;' ~ g (x,g'(x,z» a a

y~

e/;z

0-1 3)

r;">0. Then x~ admits a denPt(x,y) of class cr. provided t>4d 2 (r+l);

for three constants sity

< /;"

1;.

r;'.

moreover Pt is of class

c r in

(x,y). provided

t>8d 2 (r+l) . Proof. We will prove that 2-38 holds with 00.=21;:

the

result will then readily follow from 2-37, and 2-27-(ii) or 2-28-(ii)

(take 8=t in those). In fact, we will pro-

ve 2-38, with the infimum being taken on

E'=($ 00)

a

'

to be chosen,

for

instead of E': because of 2-34 Il this is of no consequence. For simplicity,

set V (x):=Dty (x,O), and w:= a. a

LEVY KERNELS SUPPORTED BY SMOOTH CURVES

sup x,a Iv a (x) I, which is finite. We have seen in limzt~Dzca(x,z)=O,

that (A-2)

23 ~8

and the same argument (using

indeed yields that -1

Hence if

f,

z~4>l

:0-

o

=(I+D c) x

a

lim t sup ID c (x,z) 1=0. z x z a - I, there is a $1>0 such that 00

Ull (x z)D < _1 IT Al a' - 4~ v2A/\4

(3-14)

where I.U denotes the operator norm. Let U

a

be as in

the proof of 3-8. Then 3-7 yields

+ yT(I+ll (x,z»)[Dty (x,g'(x,z)-V (x)]}. a

Ct

0.

Thus 3-14 and (D) yield for 4>2 = $l/\p(i Ya(x)=O,z~4>2

lyTUa(x,z) I

(3-15)

~ ga(x,g~(x,z»-1 d

.

{lyTVa(x)l-tlyl

yTB(x)y<e; ly12/2

If

at least one a with ya(x)=O and ) •

~):

:0-

Now we fix x, yElR by (S B '

a

M}.

there is

lyTVa(x) 12~ElyI2/2A,

Thus 3-15 results in

and 3-13 yields, i f $=~' A4>2:

z~4>:ofor some x, yElR

d

l y T Ua (x,z)1 2 0,

if

~

e; e- 2 l;z lyl2 8Al';,,2

yT B (x)y<elyI2/ 2 • Therefore, for all

,

~

to /\4AI;" /1.1_

2

).1 y 12 •

So we have 2-38 and the proof is finished.

24

3-16 think

MALLIA VIN CALCULUS

REMARK: 3-13 looks strange, as one would rather that the bigger the density is,

the more regula-

rity one gets (because a "big" density means that there are many jumps). But in fact,

3-13 does not say fIg

not too big", but rather " g (l is regular enough".

a

is

CHAPTER II TECHNIQUES

We are going in this chapter to introduce two different tools which we need later. The first one consists in a systematic exploitation of ideas of Malliavin [19]

to the end of proving exis-

tence and smoothness of the density for the law of a mutidimensional random variable.

It takes the form of

an abstract formalism which will perhaps appear as mere nonsense:

its sole interest lies in the fact

that it

avoids repeating twice the same argument, for Bismut's and for Malliavin-Stroock's approach. The second tool consists in various stability and differentiability results for stochastic differential equations depending on a parameter. These results are more or less well known, although the presence of a Poisson measure complicates the matter (in fact,

essen-

tially the statements, because the proofs are the same as for the Wiener component).

Naturally, we do not pro-

vide a complete theory, but we give only the strictly needed results:

so the interest of Section 5 indeed

lies in the fact that the results are given under the exactly suitable form.

Section 4: TOWARD EXISTENCE AND SMOOTHNESS OF THE DENSITY FOR A RANDOM VARIABLE

Consider a d-dimensional random variable (n,~,p).

~=(~

Our basic criterion for the law of

a density is due to Malliavin [19]

)i
and is a very simp-

le application of Fourier analysis; first part of the following lemma

~

i

it constitutes the

(see also Ikeda and

Watanabe [12]), while the second part is a refinement due to Kosuoka and Stroock [16]. This lemma can also be considered as a variant of Sobolev's lemma. 4-1 such

LEMMA: a)

If there are two c:c;nstants C>O and nElN-Jt that for ail fECoo(R d ) and every multi-index a with o

<

C I fD

(4 -2)

00

then: (i) if n=l, tfl admits a density; (ii) if n~d+1, ~ admits a density of class Cn-d-1 . b) Let nE]. .I*. If for every multi-index a with lal.:sn there is a random variable ~ ELd+E where E>O, suah CI • that for all fECoo(R d ) o

(4 -3)

then

tfl

admits a density of class Cn-l .

The only sensible way to get 4-2 seems to be pro1 ving 4- 3 for some ~aEL : so, going from (a) to (b), we gain on the order of differentiability (from

26

n-d-l

to

DENSITY FOR A RANDOM VARIABLE

27

n-1), and we lose on the integrability assumption on to Ld+e:). 'l' (from L1 a

The question now is: how to get 4-3? the idea is of course to derive 4-3 for

101=1, and then to iterate the

formula so obtained. Obviously 4-3 for

1<x1=1 is an "in-

tegration-by-parts" formula. §4-a. INTEGRATION-BY-PARTS SETTING In this subsection, we fix the r.v.

¢=(¢

1,

•• ,¢

d ) of

interest. 4-4

DEFINITION: An

integration-by-pa~ts

setting for ¢

is:

with ( iii) a linear space H~ of random variables, contained in n L P , and which is stable under C2 : that is, P 1 P <; n FEC 2p (Rn) , then F ('Y) EH ~ j i f '1'=('1' , •• ,'1' )E(H~) and (iv)

d

linear maps

'l'=('l'1, .. ,'l'n)E(H~)n

:5~(Fo'l') = ...

and

o~:

...

LP n H¢ p
I

(4-5)

I...li...('I')oj(,!,i); aX i ¢

i

2 d (v) for every fECp(R ) and every E[ 'l'

such that i f

aOAF/¢)O'!j]

=

E[

'l'EH~,

f(~){'I'Y~+lii('l')}].

we have (4-6)

At this point, it would seem simpler to merely ini troduce the sum G~('l') i = i 'l'y~+Ii~('l'): the reason for singling out two distinct terms will become evident when we start iterating 4-6. For the moment, we observe that 4-6 naturally leads to 4-3 for

1<x1=1. Indeed, suppose that O'~ is invertible

28

MALLIA YIN CALCULUS - 1

and that the entries of its inverse 0t!> all belong to . . 1 i' H.p' Applying 4-~ to ~~J=(o- ) J and to the ith compo<;' at!> ki nent [D x f(t!»04>] ~ and summing up on f ( t!> ) 0 t!> L k
ax

E(~£ Xj

E[£(4))

(4)))

L Gtj>«cr;l)ij)i], i

which is 4- 3 for

1 ex

1=1.

A less crude way of getting rid of the "nuisance" term crt!> in 4-6 is proposed in the next theorem (essentially due to Malliavin [19]

and Shigekawa [23]):

i" THEOREM: Assume 4-4 and that 0tj>JEHt!> for all i,j~d.

4-7

If PI denotes the restriction of the measure P to the set ~here the matrix at!> is invertible, the la~ of t!> -1 d under PI (i.e., the image Plot!> of PI on ~ ) admits a density on lR d In particular, if at!> is P-a.s. invertible, the la~ d of t!> under P admits a density on lR • Proof. Consider the following func t ion .p: lRd®~d-+lRd~~d , which clearly is of class C~:

J x -1

=lo

$(x)

exp-det (x)

-2

i f det(x)"'O

(4 -8) i f det(x)=O.

By 4-4-(iii), .pij (0t!»

belongs to H4>' Applying 4-6 to

i' . . i ~ J=$~J(a¢) and [Dxf(t!»at!>]

a

E[kf(4)) exp-det(0tj»

-2

and summing up on i, we get ] = E(f(4»Z.)

J

where

(4-9)

J

i .

Zj=Li($ J(04)))

i

the finite measure defined by

1

belongs to P'(dw)

L. If p' is

=

P(dw).exp-det(a~(w»-2, 4-9 yields IE'(--O-f(4»)1
with

ax.

-

c:=sUPj~dE(IZj I). Then Lemma 4-1-a Implies that

""

29

DENSITY FOR A RANDOM VARJABLE ~

the law of p1~pl

under pI admits a density; furthermore

by construction, so the first claim follows.

The

last claim is then obvious. Before deriving other consequences, let us emphazise that a good part of what follows is concerned with obtaining an integration-by-parts setting for the variables of interest, namely ~=x~ for tE(O,T] is the solution to

where XX

1-4. This is where the two methods

diverge: 1) In Bismut's method,

the construction 4-4 stron-

gly depends upon the solution itself; 2)

In Malliavin-Stroock's method, H~

the construction

does not depend on

~

4-4 is more intrinsic,

that is

(for "reasonable" ~'s,

including the X~ above-mention-

ned) • §4-b. ITERATION OF THE INTEGRATION-BY-PARTS FORMULA Again

~

is fixed here, and to simplify the notation we

drop the subsript setting for ~,

"~"

in our given integration-by-parts

thus writing (cr,y,H,c).

Iterating 4-6 means applying 4-6 again to :=~y+c(~),

~'=G(~)

which should then be in H for all V that are

needed. This motivates the present inductive construction of families of random variables:

Co is a finite set of JR.-valued r. v. , ini' eluding yi and cr J for i,j~d C

C U {o i (~) : i~d, ,¥EC r} r

r+l

For each

r

i f C CH.

r

)

(4 -1 0)

such that C r is well-defined, we write Yr for the multidimensional r.v. whose components are suc-

30

MALLIA VIN CALCULUS

ceSSive1y all the

~-va1ued

r.v.

that constitute

and we call Er the space in which Yr thus if

Cr ,

takes its values:

Co contains a(O) distinct real reve (a(O)~d+d2)

then (4-11) and the dimension a(r) of Er is a(r)=a(O)(l+d)r. At first,

we derive an integration-by-parts formula

which is more tractable than 4-6. The func tion g (x) =det (x): ial; that

lRd®~d ... lR is a polynomh:lRd~~d ... lRd®]Rd such

there is another polynomial x- 1 =h(x)/g(x)

whenever

x

is invertible.

Assume that CR.cH for some tElN,

and let <12.2. Let FECq(E) (i.e. a function of class C q on E., whose dep R. .. rivatives of all order have at most polynomial growth). We observe that g(a),

h(a),

y,

.sj(goa),

oj(hjioa)

are

polynomial functions of the components of Y t + 1 (use 4-5 for the last two functions); using 4-5 once more, we . . . 1 see that OJ(FoYR.) = GJ (Yi,+l)' where GJEC~- (Ei,+l)' Hence the formula

lI~(YR.+1)

=

Lj
"i + g(a)oJ(h J oa)]

(4-12)

+ oj(FoYi,)g(a)hji(v)} de~ines

i q-1 a function lIFEC p (Ei,+l)' and we set AF=

l. (A F ) 1 <"
i"

"i

N~w-:- i f ,¥"~=g(a)hJ

det(a)2

(a- 1 )Jl. F(Y t ),

i"

(a)F(Y.t)' we have'!' J= and 4-5 and 4-12 yield

I"< ['I'i j y j + oj ('I'ij)] = AFi (y o + 1 )' Therefore applying 4-6 J_d i" .. for'!' J and. summing on j gives (4-13)

DENSITY FOR A RANDOM VARIABLE

31

This is our basic integration-by-parts formula, which we can now iterate. LEMMA: Set Q"'det(o). Let nElN~, tElN, iElN, s~O, and assume that Cn+..,• lcH and that Q- 1EL 2n + s +£ £>0, (which implicitely means that Q~O a.s.). Por each n i 1 FEC + + (E ) there is a function AF whose compop R. ,n, s 1 t i+l ( ) . th 4 o " t b nen s evong 0 C E. • sucn av Jor a 7..7..

4- 14

..,+n

p

fEC n + 1 (R d ): p

Proof. mElN.

a)

We firstly prove the claims for n"'l.

jElN

some e:>0.

~or

m

p

0,2$,21,

Let w>O,

and assume that C cH and that Q-I EL 2+w+e: . +1 m Let FEe J (E), and $ECmb(R) be such that $(z)"'O for

Iz'-~t,

Ijl(z)"'l

Izl~1.

for

Let qEJN It • We shall apply 4-13 to the function F E i+l -w-2 q C (E) which meets F (Y )"'F(Y )g(o) Ijl[qg(o)]: i f p m q m m
'" O(FoYm)g(o)

-w-2

$[ qg(o)]

-w-3 (w+2)F(Y m)o(goo)g(o) $[qg(o)] -w-2 + F(Ym)O(gocr)g(o) q $'[ qg(o)],

-

and so 4-12 yields i fI F

( Ym + 1)

q

i L. j
'"

+ F(Y

{Q

-w-2

q Q ) ( F (Y m ) h

)oj(hjioo)g(o)

m

+

j i

(a) y

j

g (a)

Ilj(FoY )hji(cr)g(o)

m

-(w+l)F(Y)oj(gocr)hji(o)] m

+ F(Y )llj(goO)hji(cr) q
Let q+oo.

The properties of


and the assumption

32

MALLIAVIN CALCULUS

Q- 1 EL 2 + w+ E imply that q$'(qQ)Q-w-l ~ 0

Q-w-2$(qQ) ~ Q-w-2 and that

in L 1 +£, while all other terms in the

i (Y L P . Hence 6 F 1) converq m+ ges in L1 to the random variable 6F 1 (Y 1)Q-w-2, . . , ,w m+ 1 J where 6 F 1 Ee (E 1) satisfies , ,w p m+ previous formula are in n

i

p
.i

to F, 1 , w (Y m+ 1) '"

Lj..:::d {[ F (Y m) h J

. ( cr) y J

+

F(Y )6 j (h j i oa) + 6 j (!,oY )hji(O')]g(o) m m

-

(w+ 1 ) F (Y m ) 6 j (g 0

Similarly, converges in L

if 1

0' )

h j i ( cr) }.

fEC~(Rd), x f(~)Q

to D

(4-16)

-w

then

Dxf(~)Q-w~(qQ)F(Ym)

F(Y) as qt oo • Since by 4-13

m

letting qtCD yields E[D b)

x

f(~)Q-wF(Y)] '" E[f(~)Q-W-26F 1 (Y +1)]' m , ,w m

Now,

this formula is easily iterated. Let

(4-17) n,£,

i,s,F,f be as in the statement of our lemma. We start applying 4-17 with Dm- 1 f instead of f , and w=s, xm-1 m-2 m"'£, j"'n+i+1, FO"'F. Then we apply 4-17 with D 2f x mand w"'s+2, m"'£+1, j"'n+i, F 1 "'6 F 1 . Then we apply 4-17 0, ,s m 3 with n - 3 f and w"'s+4, m"'£+2, j"'n+i-1, F 2 "'6 F 1 2' x ml ' ,s+ and so on ... we end up with a function Fn= 6

which we call l!.F and whose compoF n _ 1 ,l,s+2n-2' i+1 ,n,s nents belong to Cp (En ,,+n ), and which satisfies 4-15. 4-18

REMARK:

The formula giving 6 F . , obtained by ,n,s iteration of 4-16, is quite complicated. However, an

induction on the number of iterations easily gives the

DENSITY FOR A RANDOM VARIABLE following property:

33

if

k 10 kF(yR,)1 ~ z;;o+IYR,1

e

k=O, 1, .. ,n+i+l

),

x

(YR.

is the generic point of ER.)'

tants z;;',

a'

then there are cons-

depending on z;;,a,R.,n,i,s

(but not on

F)

such that k 10 k(llF x

' n, s

)(YR.)I ~ z;;'(l+lyR. 1 +m +m

e' ),

k=O,I, •• ,i+l.

Now we easily deduce the following criterion: 4-19 THEOREM: Assume 4-4 and let Q=det(o). Define C i .. r by 4-10 lJith Co={y ,01J: i,j~d}. In order that 4> admits a density of alass c r , it SUffices that one of the following conditions holds: (i)

C

d and r+ cH

(ii) C cH and Q-1 EL 2d(r+l)+E for some £>0. r

Proof.

Assume

(i)

ly 4-15 with s=O, holds for all

first. R,=O,

The assumptions allow to app-

F=I,

for all n
Thus 4-2

lal~r+d+l, with

C:=suPO~n~r+d+l

E(/QI- 2n

/lll,n,O(Yn)l)

and the claim

follows from 4-1-a-(ii). Secondly, assume F=1 for all

n~r+l:

(ii). We apply 4-15 with s=O,

t=O,

we get 4-3 for each multi-index a

with lal~r+l, where ~a is the relevant component of (Y) Then W EL d + E ' for some £'>0, and Q- 2 1 a l II 1, lal,O lal . (l the claim follows from 4-1-b. The advantages of each one of obvious: more regularity, (i)

than in (li).

(i) or (ii)

above are

less integrability, needed in

34

MALLIAVIN CALCULUS

§4-c. JOINT SMOOTHNESS OF THE DENSITY The present two subsections should be read only when needed, and only by somebody interested in the joint regularity of Pt(x,y)

(Theorems 2-28 and 2-29).

{~x}xElRd

We suppose that

is a family of r.v.,

each one being d-dimensional. The following definitions are classical: 4-20

DEFINITIONS: a) The family

{~x} is F-continuous

F-differentiable) if the map

(resp.

x~~x

is conti-

nuous (resp. Frechet-differentiable) in L P for all pEl 1,=), at every point xEm

d

. The derivative is deno-

ted by l,7~x. b) The words "r times F-differentiable" or "r times F-continuously differentiable" are . lx x kx self-explain~ng. We denote by 1,7 ~ =1,7~ , ••• ,11 ~ , ••• a x x the successive derivatives (and 1,7 ~ =~ ) . For each xElR d

x HX , 0x) be an integrationlet (0 x ,'Y,

by-parts setting for ~x. Suppose that

{~x} is

j

times

F-differentiable; for each q=O,l, ... ,j we consider a x

finite set of real r.v. nents 0 f yx, u X , " V i",x '"

CO(q), f or

rl

including all compoTh en we d e f·~ne

o· ~~~q.

C;(q) by 4-10, starting with CO=C~(q). As above, we call yX(q) r

the multi-dimensional r.v. whose components

are all the r.v. constituting the family CX(q). r

THEOREM: with the above notation, we assume that

4-21

a} {~x} is b)

CX j

-q

j

(.q)cH x

times F-differentiable. for

cJ x"- E(IY:(q)I P )

and

l~n+q~j+l

if

~1,

1

-

<
and

l~n~j

if q=O.

p
DENSITY FOR A RANDOM VARIABLE

35

Furthermore, set QX=det(a x ). In order that ~x admit a density yM-"> p(x,y) on lR d for all xElR d , and that p(x,y) be of class c r in (x,y), it is sufficient that one of the foZZowing conditions holds: (i) j~r+2d+l and x'-'-">

E (I QX! -2r-4d-2-e:)

is ZocaZly

bounded for some E>O, or (ii) j~r+l and x~E(IQxl-4rd-4d-E) is locaZly bounded for some E>O. Proof. Let

a)

Let j=r+2d+l

..

co

d

f,fEeo(R).

in case (i),

Let n,qElli

with

j=r+l in case (ii).

l~n+q~j.

We have

E(nn f(~x»dx xn (4-22) = (-l)q fi(x)D q [E(nn fOx»]dx. xq xn Assume first that q>l. By (a), Dn f(4)x) is q times th xn F-differentiable, and its q derivative is of the form I;' n+i x x q-i+l x Ll i D i f (4))g .(V~, •• ,V 4», where g . is ~ ~q xn+ n, ~ n,~ a (vector-valued) polynomial. Recalling the definition

of'y~(q-i+l), we can write gn,i(V4>X, .. ) as G

x

• (YO (q-i+l).

n,~

Hence 4-22 equals

Jf(X)E{Ll~i~q D::!if(4)x)Gn,i(y~(q-i+l)}dX

(-l)q

(-l)q jf(X)E{L1
- -q

f(4)x)(Qx)-2(n+l) x tJ. G i O(Y n+ i(q-i+l»}dx, ,n+, i n,

where we have used 4-15 with s=O, and (b) (recall that n+ct::j, so C x +. 1 (q-i+l )CH x i f l~l~q), and the fact n

that

~-

(QX)-lE L2j+E.

we have proved that

Therefore if

MALLIAYIN CALCULUS

36

JE(nq+n F(x,~x»dx = JE(F(X,~x)Zx )dx xqyn n,q for

F

(4-23)

of the form F=f0f. If q=O, 4-15 immediately

yields 4-23 for F=f@f, with ZX =(Qx)-2n~ (yx(O». n,O 1,n,0 n Both sides of 4-23 being linear if F and continuous in

00

F

d

d

for the usual topology on C (R xR ), we deduce 00 d dO that 4-23 holds for all FECo(R xR ).

A

b) Let

~d. We

be an open bounded subset of

will apply Lemma 4-1 after replacing

P

by PA(dx,dw)=

lA(x)dxxP(dw), and 4i by (x,w)~~(X,lU)=(X,~x(w» and d by '!'

a

d=2d.

If a

is a multi-index with

is the relevant component of

lal=n+q~j

(X,W)t-7 ZX

n,q

(w)

and i f we de-

duce from 4-23 that (4-24) Now, we recall that 6 F 0 is a function with poly,r , nomial growth for F=G i or F=l. Then we easily den, duce from the assumption (c) and from (i) (resp. (ii) 1 2d+e:'that'!' EL (P A ) (resp. ¢ EL (P A ) for some e:'>0). a a _ d d Therefore Lemma 4-1 yields that the law of ~ on 1R x~ , under P A , has a density,

say PA(x,y), which is of class

Cr. Since PA (x,.) is clearly the density of ~x for almost all

x

in

we are finished

A

and since

A

is arbitrarily large,

(observe that the F-continuity of ~x

and the continuity of x~PA(x,y)

imply that PA(x,.)

is

indeed the density of ~x for all xEA).

§4-d. MORE ON JOINT SMOOTHNESS As the reader has undoubtedly noticed, section will 'be applied to XX is the solution to

~

x

~he

previous

su~

x

=X t for a given t, where

1-4. Here we want to study the

joint smoothness in (t,x,y) of the density Pt(x,y) of

DENSITY FOR A RANDOM VARlABLE x~,

However,

37

in order to avoid introducing now the ne-

cessary assumptions on the coefficients of 1-4, we will stick to our abstract setting, For each tE[to,tll

we consider a family

of d-dimensional r.v. with their integration-by-parts . x x HX x sett1ngs (Ot'Y t ' t,Ot)' The sets introduced in §4-c are now denoted by eX (q), and we also write YXt (q) for t, r ,r the multi-dimensional r.v. whose components constitute eXt

,r

Finally, we assume the following ad-hoc hypo-

(q).

thesis, 4-25

for which we need some terminology,

A family (fu)uEU of functions k-DUPG (for: ~olynomial

rentiable, and where

~,e

~ifferentiable

k times ~rowth)

d'

with Uniform

ID~tfu(X) 1~~(l+lxIS)

for

X

are constants independent of

(i) a finite measure

is said

if each fu is k times diffe-

HYPOTHESIS: Let jElN

4-26

.

lR u -+ lR

m

with

j~2.

O~t~k, u.

There exist:

on a measurabZe space

(ii) two families of functions (A~) uEU: lR d and (A~)uEU: DUPG; "') ( 'l.'l-1.

a

lR d

J...

-+

'~ am-z..t,h'

lR d0lR d , with

("')

'l'u

uEU

(A~)uEU being

"'~.L' Junc,,-z..ons: ltd R

0,)

-+

-+

(u,~);

]Rd

(j+i-3)-

:md

which

is j-DUPG, with Idet(D

x

c/J

u

(x» I .::

~

> 0,

al1 uEU,xElR d

(4-27)

Moreover, if f E C2p (Rd) and if 2

Lf (x)

L

i= 1 (here,

<. , • >

!m(du)


c/J

u

(

x) >

(4 -28)

is the scalar product on (lRd)ei), then

t~ E(f(<:>~» is differentiabZe on [ to,tlL with deriva-

38

MALLIA YIN CALCULUS E (Lf (~~)) ,

tive

we shall see later that the generator

L' of XX (see

1-3) can be written as 4-28. The form 4-28 easily allows for iterating the operator

L, as seen in the fol-

lowing lemma: LEMMA: Assume 4-26 for some j~2.

4-29

Let nEJN*' with

the rei s a f am i ZY ~ t·1..ons:1R d .... (lRd)~i W h·1..a.h '!.s . luna U 1 , .• , un U i 2 d (j+i-2n-1}DUPG, and such that for fEC n(R ),

n.-:j /2. ( B n,i

For a Zl i )

= 1 , , • . , 2 n, 0f

EU

p

2n

I

Lnf(x) =

i=l

J .. Jm(du 1 )··m(du )


n

ul,··,u n

where

q, 1=q, , U

U

( x) , (D i . f)

rpn+l u l ' •• ,u n + l

Proof. For n=1

X1

0

= q,D U

(4- 30)

$n (x) > , ul,,·,u n 0

l '.· ,un

$ un+ l '

the claim reduces to 4-26, with B 1 ,i=A i . u

u

Suppose that the claim holds for some n, with n+1~j/2. 2n+2 d n Let fEC p (R)j then L f, as given by 4-30, is of class C2 and we can apply L once more. Due to 4-28, it p n+1 is clear that L f is again of the form 4-30, with 2n+2 terms,

each one being an (n+1)-uple integral. Mo-

reover, the contribution of the ith term of 4-30 to Lo + 1 f is: _ for Bn+l,i ( •• ) u

_ for Bn+1,i+l ( .. ) u

-

Al(D Bn,i)o¢> + A2(D2 Bn,i)o$ u x u u x2 u A1(Bn,i(D rpn»orp +A 2 (B D ,i(D 2 rpn»orp u x u u 2 u x

n+l,i+2 for B( .• )ti. :

Then we easily deduce from the properties of

A~,B~:i,rpu

DENSITY FOR A RA."IDOM VARIABLE and is

~n

(which is also ul' .. ,un (j+1-2n-3)-DUPG.

4-31

(j-l)-DUPG)

39

that Bn+1,i

Let jElN~ be such that, with the nota-

THEOREM:

tion introduced before 4-25: for tE(to,t 1 ),

a)

tiab7,ej

for every bounded set A we have for a7,7,

cJ

O..::n+q~j+1:

sup

xEA,tO
p<~,

Eclyx (q)IP); t,n

d) 4-26 hoZds for j+2. Furthermore,

"let Q~=det(cr~).

In order that ~~ admit a

density Y"-Pt(x,y) on lRd for aH xElR d , tE(tO,t 1 ), which of c"lass C r in (t,x,y) on (to,tl)xlRdxlRd, it is sufficient that one the foHowing two conditions hoZds: (i) j~2r+4d+3 and, for every bounded set A there is with sup E(IQ xt l-4r-8d-8-C), or c>o xEA, to
(ii) c>O with

j~2r+1

and. for every bounded set A there is sup E E(I Qxt ,-4(r+1)(2d+l)-E), and x A, to
in 4-25.

$u(x)=x

The restriction ~u(x)=x in ( 11) is quite a serious

L' given by 1- 3,

one:

applied to the generator

that

K=O (which corresponds to a process

r

i t means

which is

continuous). Proof. (ii),

a)

We set r=r+2d+2

so that

j~2~-1

in case

(i), i' =r+l

in case

and that for every bounded set

A

there is c>O with (4 - 32)

MALLIA YIN CALCULUS

40

2.

2.

R,

0R,

For each R,>1 we set (U ,Y ,m )=(U,ll,m) ,and we also o 0- 0 0 0 0 set (U ,~ ,m )=({O},P({O}),E;O); set 1, vEU . Thus in view of 4-30, v 0 and with the convention L f=f, we have 2n

I Jmn ( d v)

L n f (x) =

i=O


n i i n ' ( x) , (D i f) c

, x v

v

for b)

Let k,q,nElN

'"

C o « t O ,t 1

»,

with l-.:::k+q+n-.:::r. Let '" fE

be fixed, -

'"

d

f,fECo(R).

(4-33)

O-.:::n-.:::i'.

Using 4-26 and an integration

by parts on parts on

(t o ,t 1 ), then 4-33, then an integration by d m , we obtain (the various summations of com-

ponents are straightforward,

and left to the reader:

there seems to be already enough indices coming from order of differentiation!):

f t

tl

o

dt[nkkf(t) Dq f(x) t xq

E(D n f(~x»dx xn t

tl (_l)k [ dt [let) Dq f(x) to xq (_1)k ftldt/dx f(t) t

k (-1)

o

(4-34)

EUk(Dnnf)(I:)}dx x

Dq [(x) xq

t1 2 k +q / dtJ dx f(t) f(x) mkCdv)Pi(t,x,v) to i=O uk

I

/

where

Suppose first

that

~1.

There are polynomials

(for s-.:::q) such that for every function then vqh(.Xt)~Ll

<s
DS

XS

h

gs

of class cq,

h(lx)gs(yxt O(q-s+l». Hence t ,

DENSITY FOR A RANDOM VARIABLE

41

~ DR." {(DD+i f)o
where the family

is clearly (2f+i-2k-s+1)v v DUPG. Of course Di(t,x,v) is the expected value of this x x expression. For R.,s as above, we have e t , 1-1 (q-s+1)cH t and l
where,

in virtue of 4-18,

(2r+i-2k-s)-DUPG.

"i s R. the family {G v ' , }v is

For q=O we have the same eXpression,

~1,O,O(yX

with only one term

v

t,O

(O»=Bk,i(~x). v

t

Next, for each 9=1, ... ,j+2 there are polynomials h

s,

e

with

e 5 k k 9-s+1. k L {(D f)o
5=1 where for 5=9,

k 08 • Due to 4-27, v

h9 e(···)=[D
X

thus invert the triangular system above,

we can

so that

s (D s f)o¢k = L {De (fo¢k)}h 5 ,B x5 v 6= 1 x 6 v v

where, due again to 4-26-(11i), (2r- 5 +6)-DUPG. Hence

the family {h s ,6}

v

v

is

42

MALLIA VIN CALCULUS

For e,~ as above, we have C~,e+~_1 (q-s+1)cH~ if q~l, and

C~ , 8-1 (o)cH~

i f q=O,

while

E(

IQ~I-2R.-28-e)


Applying again Lemma 4-14 yields ~ 2..

~

n~i

t=O

s=tVI

8=1

P.(t,x,v) = 1.

L

L

E{fo.p~(~~) (Qx)-H-26 t

Hi ,s,l,8(yX

( - +1»} t,l+e q s

v

where the family {H i ,s,t,8} is (at least) (O)-DUPG (if v v q=O there is only the sum over 8, and one reads Hi ,O,O,8(yx (0»). Therefore if v t, 9

Yr r

i=O t=O s=ty 1 R i ,s,t,8(yX ( - +1» v t,t+6 q s we see that 4-34 equals

I tl

dt

I dx '"f(t) -f(x)

E{

to

2n

I 1m k (dv)

x Zt(v)

k x fo4>v(~t)}'

i=O

In other words, we have dt

Jdx

I

=

k+q+n x E{D k nF(t,x'~t)} t xqy (4-36) t! dt dx E{ mk (dv)Zt(v)F(t,x,.pv(~t»)}t x k x

I

I

to for all F=fSf0f and, by linearity and continuity it imco

d

d

mediately extends to all FECo«tO,tl)xR xR ). c) Let

A

be an open bounded subset of

~d.

We

will apply Lemma 4-1 to PACdt,dx,dw)= x ICt t )(t)dt x I (x)dx X P(dw), and '"~(t,x,w)=Ct,x'~tCw», and

0'

A

1 ""

co

d=2d+l.

d

d

Then for FECo«tO,tl)xR XR ), 4-36

reads

'E

A

(D k + q + n F k q n t

x y

I

(i» tl

to

dt

Jdx

A

k (4-37) E{I m (dv)Z~(V)F(t,X.~~»}. Uk

DENSITY FOR A RANDOM VARIABLE

Now, 4-32 and {Hi,s,i,e} v

v

43

(c) and the (O)-DUPG property of

and 4-35 imply that

sUPxEA vEUk t
This being true for all

k+q+n~f

we deduce the result

under hypothesis (i), from Lemma 4-1-a. d)

It remains to prove the result under (ii). This

hypothesis clearly implies that x Id+ E ' ) < '" sUPxEA vEUk t O (recall that d=2d+1). Then if

and if fm

k

~

/u/=k+q+n

is the relevant component of (t,x,w)

i (dv)Zt(v,w),

~~

k using now ~v(x)=x. we deduce from 4-37

that EA(DUF(i» and since

~uEL

=

EA(F(~)~u)

d+E'

~ (P A)

the result follows from Lemma

4-1-b (same argument as at the end of the proof of 4-21).

Section 5: STABILITY FOR STOCHASTIC DIFFERENTIAL EQUATIONS

In this section,

the basic ingredient is a filtered

probability space Ul,r;'(~t)tE[O,T] ,P)

endowed with

a) an m-dimensional Brownian motion

W~(Wi)i

<m

'

b) a Poisson random measure ~ on [O,T] xE, where

(E,&) is some arbitrary measurable space. We suppose ~

that the intensity measure of where

G

as usual

is v(dt,dx)=dtxG(dx),

is a positive a-finite measure on

(E,~),

and

iI~\.I-v.

These assumptions implies in particular that W and

~

are independent. The L P estimates needed to establish existence, uniqueness and stability or differentiability are all consequences of the following lemma: LEMMA: Let nEn 2 < p
5-1

P

also on

T,

n, m, d) such that t

IP)ds

if

H

is an

d lR -

b) E(IKlI-wlot- p ) < 0 J E(IK IP)ds t P 0 s va"lued predictab"le process;

if

K

is an

d m lR elR -

a)

E(IHHj""P) t

va"lued process;

< 0

P

J0

E(IH 5

t

..

d

t

.2 0p J E(L~)ds if U is an lR -valued ~0~-measurab"le fun8tion on nx[O,T]XE and L is a oj

E( IU-IfiJl / )

predictable process with IU(w,s,z) I ~ Ls(w)n(z).

44

STOCHASTIC DIFFERENTIAL EQUATIONS 5-2

REMARKS:

45

1) In particular, if the right-hand side

of these inequalities is finite,

the corresponding sto-

chastic integrals on the left side exist. 2) On could actually get of the dimensions

d

and

op

independent

m.

3) This lemma is proved in [5]

for

p

of

the form p=2r, rE1N*, and we really need it later only for such p's. However, this is trivial for

it is true for all

pE[2,,,,,):

(a), and a trivial consequence of

Davis-Burkholder-Gundy inequalities for

(b);

for

(c)

it

needs an interpolation argument as the one given in [2] for example.

§5-a. GRADED STOCHASTIC EQUATIONS We consider equations of the form

where

H

is an

]Rd-valued cadlag process

(the "initial

condition") and

~e

A:

]Rd xnx [ 0, T)

~

lR d

B:

]RdxQx[O,T]

-+

lRde]Rm is

C:

]Rd xnx [ 0, T] xE

-+

lR d

is

is

~

d

0~-measurable

~

d

®~-measurable

}O_4 J

!ds~s~-measurable.

shall need coefficients not globally Lipschitz, but

.... hich have a "Lipschitz lower triangular structure" already encountered by Stroock [26)

(for the very same

reasons). This motivates the following: 5-5 DEFINITION: a) A grading of ]Rd is a decomposition lR d = ]Rd 1 x ••• x lR d q with d 1+ •• +d q =d. The coordinates of

46

MALLIAVTN CALCULUS

~d are always arranged in an increasing or-

a point in

~

der along the subspaces ~f ~ = d

1+ •• + d R. for b)

i

MO=O

the coefficients (A,B,C) d dl d the grading ~ = lR x ••• xlR q i

i'

(

only through the coordinates

5-6 dl

REMARK: Call d

x •• • x~ R.,

and

I..,:: R...,::q •

(y,w,t) and B J(y,w,t) and C (y,w,t,z)

y

lR

,and we set

\, e say that

according to A

di

in

ITa

y

k)

if

depend upon

1
-

are graded

-

when r

the orthogonal projection on

~

(ITO=O).

Then

(b)

above can be wri-

tten

( 5-7) and the same for

B' k

and

C.

Of course,

since 4- 3

is "coordinate-free" one could equivalently define a grading as being an arbitrary increasing sequence (ITO=O,TI 1""

.TIq::I)

of orthogonal projections. Then

A

would be graded if 5-7 hold. The inequalities 5-1 lead us to define the following norm,

II~IJI

'p

for processes: T

=

{j 0

E(IZ

s

IP)ds} 1/p

(5-8)

HYPOTHESES ON THE COEFFICIENTS: (A,B,C) are graded d d according to lR Ix x~ q. Moreover 'let nEn2 ..,::p
and if F is anyone of the functions F(y,w,t,z) = C(y,w,t,z) ' A( y,w,t ) , or B ·k( y,w.t ) , or () " th en F1.,S differentiable in y on (i) (ii)

d

lR

and

n z

IF(O,w,t,z) I ..,:: Zt(w) IDyF(y,w,t,z)1

~ ~t(w)(l+lyle)

STOCHASTIC DIFFERENTIAL EQUATIONS

la~-:-Fi(y,w,t,Z) I .:: r;

(iii)

i f Mr _ 1.::i,j'::M r

47 for

J r;,6~0

where

cesses such that

pEl

A

z,z

are constants, and

,,~,,~

,,~,,~

and

are predictable proare finite for all

1,00) •

THEORE~: Assume 5-9 and also that

5-10

IHI*TEn

LP. P <'" Then 5-3 admits a unique solution Y (up to evanescent

and for every pE ll,oo) there are constants c

sets),

p

and Yp depending onZy upon (r;,e'{"~"PrEl 1,"'»' such that

Ilizil ~ ).

+

(5-11)

p

Proof. a)

If the grading is trivial (Le.

q=l in 5-5)

the coefficients are globally Lipschitz; existence and uniqueness and also

Y·En T

(see for example [5,A6]; tion

H b)

satisfying

p <'"

LP

are then well known

adding a random initial condi-

IHI*En LP T p
changes nothing).

Consider the general case. We use notation

5-6 and we set Qt=IT t -IT t _ 1 • If by 5-7

Y

satisfies 5-3,

TI~

of

then

ITtH + ITtA(ITtY_) .. t (5-12) + ntB(TItY_)~W + ITtC(TItY_)*P. Due to

(a).

IITlyj*En T

P <'"

5- 12 for t= 1 has a unique solution 111 Y.

We proceed by induction on t. t~

1,

and

LP •

Suppose that for some

5- 12 has a unique solution ITtY and that

lIT yl·en L P • Then X t T P <'" if and only if n£x=ITtY

is a solution to 5-12 for and

£+1

fl..L-\LLJA VI'.; CALCLLUS

l(x,~,t)=Q.'i.. + lA(n'''t (w)+G,X. + lx),~,t) and simii~ i and ?ro~ 5-9-(iii) we deduce thac larly for

E.

(l,i,~) are globally lipschitz, c

q

.J-,-

\., l' ,.

.

TZ:llp<'"

satisfies

clearly

~ii""I"'T~-' ( _ p

<~

;,.p),

for (a)

Annlving _ _ J

_

c)

It

Y,

remains to prove

again we suppose first A(i'_)

if

l..::z+:;iyJ

and

(recall

yields that 5-1j

IF

p depending on

•

estir.H!te 5-11. !lere

the grading

similarly for

is trivia!.

p

< '.:

(t)

[T;'(H*P)

p'~

p

(nct

such that

J

+

t

SO

and for C/r.. Thus

3

t

H,A,B,C)

:-. .9.5

TtU5 o~r

hence 5-3 has a

t+!:

<::::'

the

that

1,"')

(t)=E(Y· P ), Lemma 5-1 yields a constant ~

b

b

Y~En T P

and

I S + 1 ) ,n,.,1,"., '_'. c '-..

o..;; + 1"v !*E~ LF. T P <'"

induction hypothesis obtains for unique solution

they also ~eet

--~-t-

agaif'.

unique solution Q£+!X, with

a

an~

I + I- v al2. p'::[

'7" t -- '7.t + AZ t ,\':i ' .-I' , 't' t-

h ,-

\' I' ')

t

o

E(ZP)ds s (S-lt.)

I-,'e

kno\,' that

b

p

(t)

<"',

so

(r P T5 :,

p

(t)

,5

<

-

e

P

In other words,

~

Gronwall's

)

Le:nma

yields

"liz! I

P {II H"" I! P + 'F :' . T LP 1 .. P

there

is a

consta~t

~'

(depending on

p

~)

suc:-, that < ,3' {:I E

p

Now, P

+

I'IZ III' P :-.

(5-15)

set a

C;i.)=IIr:iy;1I ,and P LP have seep.. tr.a': r::'. 'jY satisfies

• we LP whose coefficients meet

(t+l)=!IQ'-'-lY'rll

5-1~,

T Lll

the general case: I!'

f

* II

.c

<,'

5-9 with

T

STOCHASTIC DlFFERENTL\!. EOCATJOr--:S .J

0:

ir.steac

Z

Z.

49

Then

5-15

yields a

p

(;+1)

< a


p

?

r

+

~

(£.)

+

.S' ':1 II "" I

Le~

COROLLARY:

.".

Z.z ani th2 same th~

(1.+1)

p

+! I

!zllP

T LP'

Then an

(0)=0.

sily deduce 5-1 J from

5-17

p

i' 'Zi II,-p a 2 p (n

+ Moreover a

(;.)

-'-

I :£i II.,

n

-~

-p (.~• + 1 ) (n

a?

induction on

(H;A,'O,C)

f~nction

functio~

F

2 allows

to

ea-

5-16.

~

~;,d

(i";A',B',C')

~nl

in i-9.

ccrres[cnding soZutions. Buppcse

ene of the

(5-16)

'.

i~

~a!Z

aZn~

8::::.;"-

Y an,: y'

that

:~r

ccrreBpcn~:~?

5-9, and the

anb ~.

(associated with A',B',C'),

IF (y, w, t, z) -F'

(y,

0;',

t, z)

-

<

Zt

(w)(1+'

'

yl e)

(5-18)

are constants c' and y' depending u;cn C:;,e,i:I'izllVr E [

lP",),~I!Y~IIL:-}rEf

II (y-y')"'"II

Proof.

1'''=1'-'['

c' (Ii (H - H ' )~, II .,' + II P L(P

<

T LP

is solution

A"(y,OJ,t)

and

[B(Y

)-B'(Y

zlll'yp,).

(5-19)

an equation 5-3 with

'I

)]"'W+[C(Y_)-C'(Y_)l*ut(S-tO)

A'(Yt_(w),w,t)-A'(Yt_(w)-y,w,t»)

=

the same for

wi th Z't'=O and

to

)lH

H"=H-H'+[A(Y_)-A' (Y t

1,,,,»' e:wl1 that

B",e". These coefficients satisfy 5-°

Z"=2 9-1'" Z (1+ 1''''8 Y )

,to.

t

t

. t -

'

50

50

MALLIA YIN CALCULUS

UY;U

1I~1I11~

< C(p,6)'(II~1112p + L2P ) Hence 5-11 yields two constants "

for a constant c, y P

It"

c(p,9).

depending only P

upon (l;,e,{II~II~}rE[1,"')'{OYT"v}rE[I,,,,», such that ~(Y-Y')*TD ~c DH;B • Moreover Lemma 5-1 applied to LP p L Yp 5-20, and 5-18, yield a constant 0 such that p

*Yp ",y T ""yp .... SY p E(H" ) < <5 {E(IH-H'I p)+f E(Z (1+lyl »dt} T -p Tot T

~

Ii P {E ( 1 H- H '

!~ y p ) + n! I~III; p +nil ~III; ~

Then we get 5-19, with y'=2y Ii (1+T I / 2y p IIY*'T D 26 ). P P P

L

YP and c'=

P

nY; nLY~ 6 YP },

P

YP

§5-b. DIFFERENTIABILITY We have already encountered F-differentiability in 4-20 for random variables. For processes it goes as follows (A is a neighbourhood of

in

0

~n). A

5-21

DEFINITIONS: a) A family (H )AEA of processes is

called F-differentiable at 0, with derivative aH= caH i ). (the derivative is also a process) if ~


LP

-

IHAI*En

-

*' H(H A-H 0 -ClH.A)TD

T

P
for all AEA, LP

I = o( I A)

as A"" 0 , for all pE[I,"')'

A

b) If CU )AEA is a family of functions on nx[ O,T] xE, the definition is identical, provided we replace IHAI; by sup luACw,s,z) I and A 0 *' A s,z A IH -H -aH.AI T by Iz I;, where Z is a predictable

process such that derivative

au

1

if(s,Z)-lP(s,z)-au(s,z).AI~z~: the

is then a function on nx[O.T]XE as well

(the predictability of ZA is an ad-hoc hypothesis, aimed to avoid repeating over and over the same assump-

51

STOCHASTIC DIFFERENTIAL EQUATIONS

t ion) , (According to our standing conventions, the derivaA tive is a row vector-valued process when H is scalarvalued, or a tensor-valued process with one more covariant index when HA is a tensor). Now, we consider a family of equations of the form 5- 3:

(5-22) indexed by a bounded neighbourhood

0

in

~n,

HYPOTHESES ON THE COEFFICIENTS: Eaah tripZe

5-23 A

A of

A

A

•

, d

do

graded aaaord1.ng to lR 1 x ..• )(~ " Moreover, let nEn2 LP(E,G), and let FA be anyone of the A 2P <ex> A A k funetion8 F (y,w,t,z) = A (y,w,t), or B ,0 (y,w,t),or CA(y,w,t,z)/n(z); we suppose that: (A

,n

,C

)

1.8

(i) FA is of aZass c 1 in y on twiae differentiable in y for A=O;

lR

d

,

,

for aZ Z ).,EA, ana

(ii) IDr FA(y,w,t,z)1 .::. Zt(w)(l+lyI9) for r=O, r=l, yr and all AEA, and also for r=2 if ).=0; ( -:0'1:) ~V_

IDr r F/..( y,w,t,z ) - Dr r FO( y,w,t,z ) I <

y

y

Zt(w)(l+lyI9)IAI

for

r=O,l;

')13 ( 1.V -a--F A,i( y,w,t,z

y.

some

r"'::q;

where

l;,e~O

)1

< 1;

if

J

are constants and Z is a prediatable pro-

cess such that Illzll~ <= for all pEl 1,00),

5-24

THEOREM: Assume 5-23 and also that

(i)

tive aH;

).

(H ».EA is F-differentiable at 0, with deriva-

52

MALLIA VIN CALCULUS

(

{A}. (

-! )

~~

0·

O)}

, B}. (

a(C/n)

AEA'

\

O)}

(also denoted

yO

y_

aC/n,

AEA'

{l

n(z)

C

A (0

)}

Y_,z AEft are F-differentiable at O~ with derivatives
Y_

n is the same as in 5-23,

is the solution to 5-22 for

1..=0, which exists

by 5-10).

Then~ the family

O~ and

(yA)AEA is F-differentiable at

its derivative 3Y is the unique solution to the linear equation obtained by formal differentiation of 5-22, namely: ay

aH + [ClA + D AO(yO)
-

(5-25)

+ [aB+D BO(yO)ay ]It"w+[ac+D CO(yO)ay ] ,.. il • y

-

-

y-

•• Proof. Each (A A,B A,C A) sat1sf1es 5-9, and

hence Y

A

exists and

Iy

A

ITEn

I H A I TEn


If"

P

It-

P <00

L P • 5-25 is a linear

equation, hence it has a unique solution aYe Set

v~=yA_yO_aY.A; the only remaining thing to prove is that u
LP A

is clear that V tial condition

0(11..1)

as A.... O, for all

p
Now,

it

satisfies an equation 5-22, with ini-

KA = \

l.1
-), -A -),

KA,i

and coefficients

(A ,B ,C ) given by:

->.

° °

A (y,w,t) H )·

DyA (Yt_(w),w,t)y, and similarly for -

HO

-

-A -A B , C ;

"H ' a • A;

IA>'(Y~)-Ao(y~)-aA.~*t+IBA(Y~)-Bo(y~)-aB.A]"'w

° ° °

+ Ic A(y_)-C (y_)-ac.A]*fl; IAA(yA)_AA(yO)_D AO(yO)(yA_yO)]*t -

-

y

-

~

+ [BA(yA)_BA(yO)_D BO(yO) (yA_yO)]*w y + I CA(yA)_CA(yO)_D CO(yO) (yA_yO)]*ii. y -

STOCHASTIC DIFFERENTIAL EQUATIONS

53

-A ,B -A ,C -" ) obviously satisfy 5-9 with Zt=O and AZt (A

Zt(l+lyOI:~) 'I; II(v " )T"

~

and

as in 5-23 and 6=1. Hence 5-11

* II (K A )T8

of ", 0(11.1)

for some

gives

independent LP P L Yp " " and thus it suffices to prove that n (K )T" < c

c p , Yp

for all P<"'. That II (K",l);"

from (1);

* R(K " ' 2 )T8

that

LP

l~llows

= 0(1,,1)

0(1,,1) LP follows

from (ii)

and an application of Lemma 5-1. Consider now SA = C"(yA)_CA(yO)_D CO(yO)(yA_yO). -

-

y

Y~(w,z)

Taylor's formula gives a function

-

-

-

such that

CA(y~ ,w,t,z) = C"(yO ,w,t,z)+D CA(yA(w,z),w,t,z)x A

°

-

(Y~_(w)-Yt_(w»,

y~_(w)

joining

Y

t-

t

and which lies on the line segment

y~_(w).

and

Hence

SA = fD C"(y")-n cO(y")] (yA_yO) y

Y

-

+ [n cO

Y

-

(Y" ) _D

CO (y 0) 1 (y" _ yO) .

Y

-

--

Therefore 5-23-(ii,iii) yields <

2 6-1 Z (1+ly t

O *6

I

t-

'1t6

+ly"l

t-

)

[1"llyA _yO 1+ly" _yO and similar majorations for

A"

t-

t-

t-

and

B"

1 + I (Y

°ten

t-

12],

instead of

CAin. Hence 5-1 yields

n< K A,

3

t T II L P

< 26- 1

+11 (yA/T eu 3 ) f L P

Finally,

0P

II IzII1 3p

f

T

I" I u (Y"_yO)~n

3 +0 L P

L3p

(Y"_YO)~D

(5-26) 23 ) . L P

5-19 applied to y" and yO yields (recall that

5-23-(1ii) holds for r=O): U(yA_yOtn

<

T Lq -

c'[D(H"-HO)*T q q

,+ LYq

I "llIlzlI''yq,),

which is 0(1,,1) (apply (i». Hence we deduce from 5-26 , 3 * that n (K'" )TH =0(1,,1), and we are finished.

LP

54

MALLIA ~ CALCULUS

§5-c. PEANO'S APPROXIMATION Now we wish to approximate the solution of 5-3,

in

which for simplicity we assume that H=y, by the solutions to the following equations: yn

=

y + An(Y:n)~t + Bn(yn )~W + Cn(yn )~p, 't' .pn

(5-27)

where (An,Bn,C n ) are as in 5-4, and if

t=O

if

k T 2- n
To simplify our notation,

(5-28)

(A~,B~,C~)=

we also write

HYPOTHESES ON THE COEFFICIENTS: Each trivle n dl d qx ••• x]R • More(A ,B ,C ) is gr>aded according to 1R n P over, let nEn2 .::.p <00 L (E, G), and if F is anyone of the fu nat ion s F n ( Y , w , t , z ) = An ( y , w, t), 0 r Bn, . k ( y , w , t), or 5-29 n

n

Cn(y,w,t,z)/n(z), then (i)

F n is differentiable in y on

IFn(O,w,t,z)1 ~ Z~(w) for

(ii) (iii)

~ Z~(w)(1+lylo) for

r-

nElN;

nElN d n,i( y,w,t,z )1 (v ) I -~--F

M

nElN.

IFn(y,w,t,Z)-F""(y,w,t,z)1 ~ Z~n(w)(l+lyle)

(iv) for

IDyFn(y,w,t,z)1

1R~

<

"y. l
(v i)

Z lIn (w) if t .

nElN

and

r

a ""

1-a--F'

i (y, w, t , z)

I

Yj

where ~,e~o are constants and

zn, z,n , z"n are predic-

table processes such that for all

P
sUPnElN

,,~n,,~
STOCHASTIC DIFFERENTIAL EQUATIONS

and

ep <00 , eP

where

= sup ElN sup T DE[ (Z"n)PIF nJ D . n t2 t =$t Lex. THEOREM: Assume 5-29 and ,,~,n,,~

5-31

55

+

0

(5-30)

as nt'" for

all pEl 1."'). Then 5-3 (with H=y) and 5-27 have unique solutions, and as nt'" for all P<"'.

(5 -3 2)

Proof. a)

(A,B,C) satisfy 5-9, so by Theorem 5-10,5-3 LP. has a unique solution Y and Y*En T

P <'"

b) In order to prove that 5-27 has a solution, we begin with the case when the grading is trivial (q=l). n

Let 1jI k = k T 2

-n

.

Wee 0 n 5 t rue t y

n

n

by in due t ion a s follow s: p

n

start with Yo=y. If Y n is known and belongs to np<=L , 1jIk n n set for I/Ik
yn = yn + f AO(ynn,s)ds t 1jI~ 1jIo 1jIk

(5-3 3)

k

+

f

t

1jI~

BOCY~n,s)dW ~k

+ s

f

t

fCnCynn,s,z)P(ds,dz). 1jI~ E 1jIk

To see that 5-33 makes sense, we first remark that by 5-29-(ii,v):

IFo(ynn,t,z)1 I/I k (recall that q=l). Hn is predictable,

Cs) ,

Eclz"nlPI F »ds s =I/Ik

(5 -34 )

56

MALLIA VlN CALCULUS 2 P- 1

<

t

J

~n

k

E(IZnI P + B lynnlP)ds s P Wk

(because of 5-30 and II~nll~ <=). Hence,

<

'"

due to Lemma 5-1,

5-33 makes sense and yn l E n LP. IjIn p
sup Then,

Iynt '.jIn
on [O,TJ c) of part

and that

lynlWTEn

P
LP •

Tn the general case, we can reproduce the proof (b)

of Theorem 5-10,

provided we replace everyn

n

n

where y_, IT~y_, Q~+lX_ by Y9 n ' n~y$n' Q~+lX~n' tain that 5-27 has a solution which satisfies lynl*T En

p
L P . Moreover,

isfy 5-33 for shows that

We ob-

any solution of 5-27 must sat-

w~
the solution to 5-27

k

is indeed unique.

We can further reproduce the proof of part

(c)

of

Theorem 5-10 as well: replace" !A(Y_) !~z+z;IY-' " by " !An(ynn) !...:::zn+z"nl y n n ! It; the same trick as in 5-34

to

obtain 5-1t, with B instead of Z;p. The rest p of the proof goes exactly the same way, with ~=Zn. The-

allows

refore we obtain the estimate 5-11, sup II!znll!
and since

for all p <"", we get

n (yn) 'ATO

Finally,

L

-n

p

< '"

for all P <0>.

(5-35)

n

Y =y -y is a solution to an equation

5-3, with

-n

n

A (y,w,t) = A(y n(w),w,t) ~t

and

-n

-n

-

n

A(Y t

(w)-y,w,t) -

the same for Band C . These coefficients satisfy

STOCHASTIC DIFFERENTIAL EQUATIONS

00 I Y nl~e ) I n n I ' and 5-9-(1) Zt(1+ Y tn-Y t t and 5-9-(iii) with ~. Hence 5-11 yields

-n Zt

5-9-(i) with ~n Zt~Zt'

with

57

~

p < c (II (ii n ); II y

II (Y n _ Y); II L

p

+

L P

11IZnlii

) Yp

for some c p , Yp • From Illz,nlllp -+ 0, 5-29-Civ) and 5-35, plus the usual application of Lemma 5-1, we deduce that ~ 0 f or a 11 p <00. S0 it rema i ns to prove II C-Hn),," n ~ T LP Il1Znll~ -+ 0 for all p
'tJ

o.

-+

P

Now, we have seen that 5-11 holds for yO; it also holds for y~_ynn' which yields 'flt

nO _ t

yn ~o

t

n

n

LP

n

Thus IIYt - Y,nD n

< c

{jt Eclz n ,Y p )ds}l/Yp.

P

n

$t

s

clearly goes to

.t LP

$ttt). We deduce that

aod this finishes the proof.

0

as nt oo (because

CHAPTER III BISMUT'S APPROACH Section 6: CALCULUS OF VARIATIONS

§6-a. THE GENERAL SETTING In this section we fix mElN~ and E, an open subset of * ~ B (BElli), with its Borel a-field ~. G(dz)=dz denotes Lebesgue measure on

(E,~).

n is the aanoniaaZ spaae of all points w=(w,n), where:

- w is a continuous function:[O,T]+~m, w(O)=O; - n is an integer-valued measure on [O,TjxE,

and Wt(w,n)=w(t),

~«w,n);.)=n(.).

Furthermore,

(F)O T is the smallest right-continuous filtration =t
(n,f) with the unique probability measure W

P

such that

is a standard m-dimensional Brownian motion;

U

is a Poisson measure with intensity

(6-1)

v(dt,dx)=dtxG(dz) W and

~

are independent,

and we write

P=~-v

for the compensated Poisson measure.

We are given a point xo in coefficients:

59

~d, and a family of

MALLIA YIN CALCULUS

60

and we consider the following equation: (6-Z)

We will obviously need some regularity assumptions on these coefficients for

the purpose of the "itera-

tion" outlined in Section 4;

these conditions will be

slightly less stringent than (A-r), and we set Cr E 1N*): 6-3

ASSUMPTION (A'-r): The coefficients (a,b,c) a~e al d x ••• xlR q (see 5-5). They are r

graded according to 1R

times differentiable, and there are two constants ~,e>o and a function

Inn a(x)l, xn

(i)

nEn z.3' <'" LP(E,G)

Inn b(x)', xn

,_(l)nn c(x,z), n z xn

< ( ,...... . .:)

IDn+nkk c(x,z), x z

1; (1 +

, x , e)

< 1; (1 + I x I e) ,

I

on a exi(x,z) <1; for xil··· xi of the functions a(x)~ or b·k(x), or

(iii)

la

such that

for

°2.n ~r ;

1
l~n.::r, c~(~»,

i f ex is one and i f

M l
I

on Ok . cl.(x,Z)'5 for 1
We always assume (A'-I) at least. Thus 6-2 has a unique solution X and X~En LP in virtue of 5-10. T

P <'"

Now we briefly describe Bismut's way of obtaining 4-4.

The idea consists in constructing, for each A belonging to a neighbourhood A of in lR d :

to

°

1)

a probability measure pA (with pO=P) equivalent

P.

Denote by GA=dpA/dP the Radon-Nikodym derivative,

CALCULUS OF VARIATIONS 2) a "shift" a A on p\(e A) - l : p • Then,

i f ZEL

1

)l

61

(with aO: identity), such that

(P),

(6-4) A

Suppose that

H

5-21)and call

(C )AEA

is F-differentiable at

the set of all r.v.

A

(Zoe )AEA is F-differentiable at

Z

0. Let

°

(see

for which 1 ~=(~

, •• ,4>

d

)

be the variable of interest, and suppose also that

~iEH. Then indeed the following is an integration-byparts setting for (1. e.

04>

cr¢

4>

(see 4-4):

Cl~i.eA

ij cr4>

ClA.

J

-ClC,

Y4>

154>('1')

:

for '¥EH

(4-5 is trivial, 6-4 at A=O, with

(6-5)

and 4-6 follows from differentiating Z='¥f(4))).

A and P aA:identity and pA=p). But the shifts 6 A should

Of course, (e.g.

}

H

H4>

-a'¥

I A=O) ,

there are many choices for 6

A

in fact be chosen so as to make cr4> invertible for 4>=X t : so one sees on 6-5 why the construction 4-4 in this method depends upon the solution

X

itself.

§6-b. THE GIRSANOV TRANSFORM First, we construct the shifts e A on

°

n,

indexed by the

elements of a neighbourhood A of in ]Rd. We shall A A A A denote by W =Wo e and)J =)Jo B the "perturbed" driving terms. The idea is that the law of with the law of

(W,)J).

A

A be equivalent

(W ,)J )

It is well known that this can

62

MALUAYIN CALCULUS

be achieved for WI. by setting:

Jtu

ij WtA = Wt + ' l ' d 1s o S 01. ds, where u=(u ) I
6-6

l'

u.A=(L j

.

u JAJ) l
o

Then W =W and the perturbation is linear in A. As we enoug~

need first derivatives only at 1.=0, this is good As for ~A,

infinitely many jumps have to be perturbed

simultaneously (for reasons apparent later); this essentially requires that the jump times of

~

be left un-

changed and that only the jump sizes be modified. We can do as follows: 6-7

~

A

=y

).

(~)

is the image of A

~(w;.)

under the map

).

(t,z)_ (t,y (w,t.z», where y (w,t,z)=z+v(w,t,z).). A i_ i I;' ij j . ( i . e ., y (w, t, z) - z + L j
~

(Recall

is the predictable a-field).

n being the canonical space, the map

A

6 :n

~

n

is entirely defined by

eA

~o

(6-8)

For technical reasons, we will need boundedness assumptions on

u

and

liary function p: E

v. First, we introduce an auxi~

[O,~).

which meets the following

(the same as 2-23):

6-9

Conditions on p:

IDrzr p IEL 1 (E , G) (11) p(z)

(i) p is of class c~, with

for all reIN. ~

0

as

z

goes to the boundary of E.

CALCULUS OF VARIAnONS

6-10

Conditions on u and v: z~v(w,t,z)

(ii)

I

! v (w, t , z) ~p (z)

and

u

(1)

63

is bounded;

is differentiable, and I Dz v (w, t, z) I ~p (z) .

Observe that 6-9 implies that p(z).,::a d(z,aE), where d(z,oE) denotes the distance of

z

to the boundary of

E, and a:=supIDzP(Z)!. Then we easily deduce from 6-10-(ii) z~y

if A is small enough, for all w,t,A,

that, A

is a bijection from E onto E.

(w,t,z)

(6-11)

The first order of business is to find a RadonNikodym derivative GA such that if pA=GA.P, then pAo (SA)-I=p.

For this, we first enter the Jacobian of

the transformation 6-11,

to wit (6-12)

(I is the identity matrix with the correct size, namely

BxB here). Y

is a polynomial in A, of degree

constant term is

S, whose

1, and whose other coefficients are

themselves polynomials in Dzv. In particular, the coefficient of Ai in the degree 1 term is div v

.i

:=

I.J_<S

_0_ j i v • oz.

(6-13)

J

In view of 6-9 and 6-10,

the previous discussion allows

to shrink further A so that for all AEA, (6-14)

k.

tant As

u

1

is bounded and pEL (E,G) and 6-14 holds,

following defines a

(real-valued) martingale:

the

MALLIA YIN CALCULUS

64

Its Doleans-Dade exponential G A= E(M A)

is the solution

to the linear equation (6-15) 6-16

G~(w).P(dw) is a probabiZity . f'~es P = p~ o(e ),,)-1 . sat~s

THEOREM: pA(dw) =

measure on ( n,& ) an d Proof. ---

a) 6-15 is a A

A

l-dimensional equation of type 5-22

A

,

A

with H =1, A =0, B (y,w,t)=-yut(w).", C (y,w,t,z)= y(yA(w,t,z)-1).

Since

u

is bounded and 6-14 holds,

these coefficients trivially fulfill 5-23 with n=p and 5-24-(i, if), with tlA=O, tlB=-u, (G A)

Theorem 5-24 yields that

tlC=div v

(due to 6-14).

is F-differentiable,

and in particular IGAI*TEn LP: hence the local martinp <= A A gale G is a martingale, and E(GT)=l. Moreover, 6-14 yields that a.s.;

therefore

sure,

equivalent to

b)

pA=G;.p

I~MAI~I/2,

thus

G~>O

is indeed a probability mea-

P.

For the second claim, we apply Girsanov's Theo-

rem for continuous martingales [ 13, (7-25a)] dom measures [13, (7- 31)]

and for ran-

as follows:

(i) Under pA, WA is a continuous local martingale starting at as

W;

0

and having the same quadratic variation A hence w is a standard m-dimensional Brownian

motion under pA. (ii)

A

Under P ,

the measure

predictable projection (or,

A

has Y .v for its dual A compensator). Since y is ~

predictable by construction, pA=yA(p) has the image yA(yA. V) of yA.V for its dual predictable projection un).

der P . Now,

6-11, 6-12 and the classical change of va-

CALCULUS OF VARIATIONS

65

riables formula for Lebesgue measure on E give for every Borel subset A of [O,TlxE: [y

T

A(y A• v) 1 (A)

J ds a

Jdz lA(s,z+v(s,z).A)yA(s,z) E

T

J a

ds Jdz lA(s,z) = v(A). E

Hence yA(yA.V)=V is the dual predictable projection of A ). A ~ under P , that is, ~ is a Poisson measure under pA, with intensity v. Thus (W (W,~)

A

,~

A

) has the same distribution under P

).

as

under P. Since P is the unique measure meeting

6-1 and since (W (eA)-l=p.

A

,~

A

)=(W,~)oe

A

,

it follows that

p\

From part (a) of the proof above, we also deduce: COROLLARY: The family

(e A)

is F-differentiable at 0, and its derivative ae is given by formal differentiation of 6-15: 6-17

(6-18)

§6-c. PERTURBATION OF THE STOCHASTIC DIFFERENTIAL EQUATION So far we have obtained 6-4, and we need now to prove that the family {Xtoe A) is F-differentiable. Formally, yA=xoe A should be solution to yA =

xo

+

a(Y~).t + b(Y~).WA+ c(y~)*(~A_V).

Of course, v is not the compensator of

~

A

(6-19)

under P, so

MALUAYIN CALCULUS

66

this equation is not of the form studied in Section 5. Nevertheless, we have: LEMMA: Under (A'-l).

6-20

yA=xoS A satisfies

Proof. We consider successively and

U=a(X_)~t,

U=b(X_)~W

U=c(X_)~ii.

a) If U=a(X )~t, b) Let ,\ -1

~tC(S)

then UoSA=a(yA).t

is trivial.

U=b(X_)~W and UA=UoS A. Since p=pA o (SA)-l and

(~t)'

classical results concerning mappings of

processes (see e.g. [13,(10-38),(10-34),(10-40)])

imply

that VA is a pA-local martingale of the form b(X oSA)*(Woe A) = b(yA)*W A . As pA~p, the stochastic in- 1 s are t h e same un - d er P an d pA , so UA = tegra A A

b (Y _)* W + b (y _) u • h·t •

A A c) Let U=c(X ).p and U =U06 . Like in (b) we obtain that u A is a pA-~urelY discontinuous local martingale,

and since P

A

~P,

U

,\

is a P-semimartingale with no con-

tinuous martingale part. Now, using 6-16 and the property GTAEn p
Z: (6-22)

Since UT~En p
or [201), and i t is also quasi-left-con-

tinuous because it jumps only when UA=M+A

where

martingale and

M A

~

jumps. Hence

is a P-purely discontinuous local is continuous with finite variation.

Identifying the jumps

~M=~UA, we get that

CALCULUS OF VARIAnONS >..

>..

c (Y _ , Y ( z ) ) "'" il

M

( see [ 1 3] ) •

It remains to identify tion to

67

A. Let

fn

be the restric-

of the function: z~lA(n-lzl)+, and

E

c (x,z)=c(x,z)f (z), U(n)=c (X )r.il,UA(n)=u(n)oe A • Exacn

n

>..

tly as above we have

n,l;,e are as in (A'-I), 1

n

-

>..

>..

(z»~jl+A(n). But if Ic n (x_,z)I'::'l;(l+lx_1 6 )n(Z)f n (Z),

U (n)=cn(Y_,y

is also c n (X_)*~-cn (X_)I'rV, >.. both integrals making sense; hence U (n) =

and nfnEL (E,G); thus

U(n)

c (Y\/'(z»*"il-c (y>",z)l!"v. We deduce that n

-

Now,

n

fn

( i , ii)

-

is Lipschitz with constant

1. Hence (A'-l)-

and 6-10 yie ld

and the right-hand side above is G-integrable. Since c n ~ c pointwise as nt~ we deduce from Lebesgue convergence theorem that for all t
Moreover, Lebesgue convergence theorem for stochastic integrals (or Lemma 5-1) also yields that I U(n)-Ul; ~ 0 IM(n)-MI;

in all LP(P)

(p<~), as well as

~ O. From 6-22 we deduce that I tf(n)-u).I~~o

in LP(P), hence IA(n)-AI; ~ 0 in LP(P). From 6-23 it >.. >.. >.. >.. >.. U =c(Y_,y (z»*"il+[ c(Y_,y (z»-c(Y_,z)]*v.

follows that

Putting together the results of (a),

(b),

(c)

yields 6-2 1. {Xo6 A}>..EA is Fdifferentiable at 0, and its derivative DX:=;A(b6 A) 1>..=0 6-24

THEOREM: Under (A'-2) the family

68

MALLIA YIN CALCULUS

is the solution to the lineap equation obtained by fopmal differentiation of 6-21: DX

Dxa(X_)DX_~t

+

Dxb(X_)DX_~W

(6-25) +

Dxc(X_)DX_~P

+ b(X_)u*t + D cex z

-

)~~.

A A Proo"f. Due to 6-20, Y =Xo 9 is the solution to an equa-

tion 5-22, with HA=xO and the coefficients A

A (y,w,t) = a(y) + b(y)ut(W)A + J[c(y,z+v(w,t,z)A)-c(y,z)]dz E B

A

(y,w,t)

(6-26)

bey)

CA(y,w,t,z) = c(y,z+v(w,t,z)A). We wish to show that the assumptions of Theorem 5-24 are met. This is trivial for 5-24-(i) (with H=O), and (AA,BA,C A) is obviously graded according to the grading d dl d lR = lR x ... xlR q. For the other conditions, it suffices to prove that they are met,

A

separately for each of

A

A

the following terms: AI=a, A2 =buA, A3 (y,w.t)= rX;ey,w,t,z)dz, where Ai(y,w,t,z)=c(y,z+v(W,t,Z)A)-

E

A

A

c(y,z), and B , and C • A

A

Due to (A'-2) and 6-10, Al and A2 and B satisfy 5-23 (with A

Zt(w)=/;+

A

I;;

A

clearly

sup p(z». The families z

{AI (X_)} and {B (X_)} are obviously F-differentiable at 0, with derivatives oA1=O, oB=O. Since ~En LP , -"T P <"" it is also easy to check that {A;(X_)} is F-differentiable at 0, with derivative oA 2 =b(X_)u. A

k~1

Consider now C • We use the notation of (A'-2), and satisfying IAI~k for all "EA. CA obviously satis-

fies (0, and (ii) for A=O, of 5-23, by (A'-2). Due to 6-10 and (A'-2)

(first (ii),

then (i),

then (iii) and

CALCULUS OF V ARIA nONS ( i v) ),

69

we have:

r=O,l;

if

< I;;k(1+lyI9)[p(z)+TJ{z)]

(6-27)

i f r=O, 1

la~.

CA,i(y,z)1 2.l;;k[p(z)+n(z)]

J

if Ms_1
for some

satisfies 5-23 with n'=p+n

s2q. (which belongs

to n 2 LP{E,G». Next we prove that the family A 2P <'" {C (X ,z)/TJ'(z)} is F-differentiable at 0, with derivative ;c/TJ'=nzc(X_)v/n" kl;;(l+lx_IB), while

6-27 yields

Ivl2P and

ICA(x_)/n'l

(A'-2) yield

2

lac/n'l 2

l;;(l+lx_IS)supz p(z), whose supremum in time belong to L P • We also have P <= (A' -2) ) :

n

(use Taylor's formula and (ii) of

and we easily deduce the desired result. It remains to consider A~. 5-23 is satisfied for O. "';\ ;\ 0 ;\=0, because A 3=0. US1ng the equality A3=C -C and 6-27 and the definition of A~, we deduce that A~ is indeed well-defined, In\A;cy,w,t)1 y

~

is once differentiable, and that 1>.11;(l+lyI9)G(p)

for r=O,l

(G(p) denotes the integral of p with respect to G). Hence 5-23-(i,ii,i1i) holds with Zt(w)=l;;kG(p).

Using

(iv)

of (A'-2) and Taylor's formula, we see that a "';\, if Ms _ 1 . I p(z)

yj

s2q. met.

hence

a

;..

Iay-A 3 , j

i

(y,w,t)1

<

1;kG(p) and 5-23-(iv) is

70

MALLIA VIN CALCULUS

A Finally we prove that the family {A 3 (X_)} is F-differentiable at 0, with derivative aA3=fDzc(x_,z)v(z)dz.

Since IA 3A(X -

IA~(X_)I .5. k1;G(P)(1+lx-,S) )1*T En p
A;(X =

we

h~ve

)-A~(X_)-dA3.A

f[ c(X_,z+v(z)A)-c(X_,z)-Dzc(X_,z)v(z)A] dz. E

Since Ivl.5.p,

(ii) of (A'-2) and Taylor's formula give

and we easily deduce the claimed result. Finally,

in view of the form of dAl'

aA 2 , aA 3 , aB,

ac, Theorem 5-24 implies that {yA} is F-differentiable at 0 and its derivative

+ D c(X )DX x -

~O

DX

+ b(X )u~t + D c(X )v~U

-

+ [JD c(X

E

Now,

has

Z

-

(6-28)

z-

,z)v(z)dz]lt-t.

the last term is equal to Dzc(X_)v*v, and

IDzC(X_,z)v(z)1

is clearly v-integrable.

the last two terms in 6-28 is

So the sum of

Dzc(X_)v~P,

and 6-28 re-

duces to 6-25.

§6-d. EXPLICIT COMPUTATION OF DX.

In this subsection, we make explicit the dependence of X

upon the initial condition, writing

xo X

for the

solution to 6-2. 6-29

THEOREM: Under (A'-2) the famiZy

{Xx}xElR d

is F-

CALCULUS OF VARIATIONS

71

diffepentiabZe at each point xE~d ~ and its depivative at x, denoted by ~xx~ is the unique soZution to the lineap equation obtained by fopmaZ differentiation of 6-2: ~Xx = I

+ D a(xx)~xx*t

x

-

-

+ D b(Xx)~Xx*w

x

-

-

(6-30)

+ D c(Xx)~xx*"jj. x -

Proof. Apply 5-24 with HX=x, and coefficients BX=b, CC=c. As is well known, one can "explicitely" compute ~xx as follows.

For each

x,

introduce the following

~d~md

-valued process: (6-31) (due to (A'-2) and Lemma 5-1, this is well defined). Then 6-30 reads as (6-32) (or,

in differential form,

d(VXx) = dK x ~X~). The solu-

tion vxx to 5-32 is the transpose of the DoZeans-Dade

exponentiaZ of the transpose of

KX.

Now, we come back to 6-25, writing initial condition is xO=x.

DX x

if the

Set (6-33)

(we will see presently that

U,v

also depend on

x).

Then 6 -25 gives DX x

HX + [(DXX)T,1I-(Kx)T]T (or:

d(DX x ) = dH x + dK x DX~ ).

(6- 34 )

This is a linear equation associated with the homo-

72

MALLIAYIN CALCULUS

geneous equation 6-32.

Its solution can be explicitely X computed in terms of H and VX X , by the usual method

of variation of parameters.

Set

)

(So KX (1)=K x and VX X (1) =VXx) . and we know [ 14]

as nt"',

Since KX is ca.dlag,

i f t..::.T n _ 1 ,

v?(n)t is invertible i f t
is not invertible i f

t-

is invertible i f

Then it is shown in [14]

TXt"" n

that

X

yxX(n)t=I

(6-35)

that,

t

1

(6-36)

t>T x , - n
f

)

since HX has finite va-

riation, I{xX(n)t {DX x

TX

n-1

V'Xx (n) -1 (I+llKx) -1 dHx} ss s

(6-37)

(and

§6-e. HIGHER DERIVATIVES We see on formula 6-25 that if we want to further differentiate DXxoSA in A, we need to differentiate uoS A and voS

).

,

so 6-10 is not sufficient. On the other hand,

as already noticed before, on the solution

XX

the shifts SA should depend

itself, and in particular on the

CALCULUS OF VARIATIONS starting point depend on

x,

x:

in other words,

and we write them

u

u

x

73 and

and

v v

x

might

•

Instead of starting with arbitrary (differentiable enoueh) uX,v x , we restrict in fact to the following form,

which is sufficient for our purposes:

6-38

Conditions on UX and v X : We have UXtCw) = f(Xx (w),VX x (w)), t-

t-

vX(w,t,z) = g(Xx (w),'VX x t-

t-

f:lRdx(lRd@lR d )

where (i)

-+

(w),z),

lRm~lRd is of class c~;

(ii) g(x,y,z) = g (x,y,z)p(z), where p satisfies d d dO B d . 00 6-9 and go : lR x (lR @lR ) xE -+ lR 3lR 1S of class C b . Upon multiplying

g

by a constant, we can and will

also assume that 6-10 is met. ply for all ZM.->supx to

n1

6-38 and 6-9 clearly im-

n,k,iElN:

,y

.=:p <00

I I D n+k+i n k ig(x,y,z)

x y z LP(E,G) •

belongs }

(6-39)

Then 6-15 defines GX '" for each xElR d ; note that the shifts the family

(6 x '")

also depend on

x

and for each

x

{GX'''}"EA is F-differentiable at 0, with de-

rivative

(6-40) while 6-33 reads as

(6-41) As seen in the previous subsection, we are led to consider processes yX indexed by xElR d • We have two

74

MALLIA YIN CALCULUS

kinds of derivatives: 6-42

x

The family {y }xE]Rd each point

Xj

may be F-differentiable at

the derivative is then denoted by

Vyx. If one can iterate the procedure one says that

yX

is

r

times,

times F-differentiabZe

r

(in x) and we denote by VkyX the successive derivatives (with vOyx=yx). 6-43

The family {yXoeX,A}AEA may be F-differentiable at 0, for all Xj

the derivative is then denoted

by DYx. If we can iterate the procedure we say that

is F-(r)-differentiable

yX

and we denote by

Dkyx

times

(in A)

the successive derivati-

ves (with DOyx=yx)j for instance, derivative of

r

D2 y x

is the

{(DYx)o eX,A}AEA.

Our aim in this subsection is to prove: 6-44 THEOREM: Assume (A'-r) for iome r~3, and 6-38.

Then a) {Xx} is (r-l) times F-differentiable in x, and {aG x } is (r-2) times F-differentiable in x.

0~k~r-2,

b) If

{Vk~} is F- (r-k-l )-differentia0~k~r-3. {vkaG x } is F-(r-k-2)-

ble (in >.: see 6-43)jif differentiable (in A).

Furthermore, VkDnXx = Dnvk~ for n+k
F = lRdx(lRd@lRd)x(lRdelRd)x]Rd. A point

denoted by sition of

x

of

F

is

i=(x,y,w,y) according to the above decompoF

into four factors.

6-2, 6-25 (or, rather,

If we put together

6-28), 6-30 and 6-40, we see

that XX is solution to an equation 6-2, with dimension,

CALCULUS OF V ARIA nONS

75

d=2d+2d 2 , with initial condition x~(x,I.O.O). and with the following coefficients:a,b,c: -

i f i.3l:

-i

a

(x,y,w,y)~a

i

-in in (x), b (x,y,w,y)~b (x),

ci«x,y,w.y),z)~ci(x,z). -

if d
2

dinate in 1R -i

and d

i

corresponds to the (j. k)

®:m. :

a (x,y,w,y) =

a·

\

---aJ(x)y

Ln
"

-

c

«

_

x , y , w , y) , z ) -

ax-b 0

jn

,

(x)y

R,k

,

R,

\

l. i


_0_ j d x c (x, z) y

-

-

coor-

R,k

d x R,

-in \ b (x,y,w,y) = l.R,
th

d

R,k.

,

i

if d+d 2
aiR,k

-~--c

(x, z) g

(x , y , z ).

oZR,

-i

a (x,y,w,y) = 0,

-i c « x, y, w, y) , z) ~

-in b (x,y,w,y)

a

Li _<13 az-g i

-f

nj

(x,y).

R,.

J (x, y, z) •

We wish to prove that (a,b,c) satisfy (A'-(r-l». Our first task is to find a grading for F=JR

d • We do as

follows: We first decompose the first factor lR dl dq to JR x; .• xlR

[aJ

[a1

d

according

Next, we decompose the second factor lRdePlR d by ar-

MALLIAVIN CALCULUS

76

ranging the coordinates (i,j) . . d into q(q+l)/2 blocks ~,

as follows:

first

{(i,j):

J <

i,j~MIT;

then

{(i,j):i~Ml<j~M2 or

j::M 1
[y]

::M 3}'

and soon •..

Next, we decompose the third factor lR d 0lR d as abo-

ve in [BJ.

[ cJ

d

Finally we decompose the fourth factor lR as in [a]. d dl dSo we have lR = lR x ••• xlR q with q=2q+q(q+1). It

is then tedious, but pretty obvious,

(;,b,c)

satisfies (A'-(r-l»

(A'-r) for

to check that

(use 6-38 and 6-39 and

(a,b,c», with 6+1 instead of 6.

-x

Then DX

exists by 6-24, and

-x

~X

exists by 6-29.

Moreover, writing 6-25 for the relevant components of DX x to get DVX x or 6-~ for the relevant components of vXx to get VDXx, give the same equation: so DVXx=VDX x . 2 x -x -x -x Now we can iterate the procedure: Z ' =(X ,VX ,DX ) satisfies an equation 6-2 with (A'-(r-2», then Z3,x= 2 x 2 x 2 x (Z ' ,VZ ' ,DZ ' ) satisfies 6-2 with (A'-(r-3», ... , and we stop at

zr-2,x,

thus getting the result.

§6-f. INTEGRATION-BY-PARTS SETTING FOR As already pointed out in §6-a, Bismut's approach provides a natural integration-by-parts setting for every r.v.

$ that is F-(l)-differentiable (see 6-43). We resx trict our attention to $=X t . ,

d

PROPOSITION: Assume (A'-2). For> aZZ xElR ~ t::T. x x x x the foZZowing ter>m (at,yt,Ht,Ot) is an integration-by6-45

Parts setting (see 4-4)

~or>

J

Xx. t .

CALCULUS OF V ARIA TIONS

DX~,

Y~

77

I

-aG~,

the set of all £X-measurable r.v. such that tiable at

o~('!') =

-Cl 'I'x ,

{'fo eX,

I

~

}), is F-differen-

\ (6-46)

0,

\

derivative at 0 of

{'!'oSX'''}),EA' X n

Proof. A standard argument shows that if '!'E(H t ) aDd fEC2(RD), then f('I')EH~, and the derivative af('I')x of p x), . xi' ix {f ( '!' ) e ' } a t 0 1 S Of ( ~ ) =L i
P-martingale show that

E[f(X~oex'A) '!'oex,A G~'),j

EX'),[feX~oeX'),)

=

'foeX'''j

= E[feX~)'¥l,

for f EC2 (R d ). Differentiating this in ), at ),=0 yields p x 0 4-6 (recall that G t ' =1). In order to apply the results of Section 4, we still have to describe the sets it. To do this, for each

j

eXt, o(q)

introduced in

such that (A'-(j+1»

holds

we set:

and for

{

eXX,DXX,aG X )

if j =0

skx sk x ({D V X }O..::s+k..::j' {D V aG }O..::s+k"::j-l)

i f j> 1

eXt, o(q)

we take thp. family of all components

of zq,x:

it contains in particular the components of

X at,

1 X

X

Yt

t

X

.

Yt , V Xt for O..::i..::q, as was required. Then, if ~(q) is the r.v. whose components constitute the set

e~', r.(q) 6-47

(see §4-b), we get:

LEMMA: Assume (A'-r) for some r> 3 and 6-38. Then

MALLIA VIN CALCULUS

78

{X~} is (r-l) times F-differentiable.

a} b)

2 (q)cH Xt for 1
eX

eX

t,r-

3(O)CH Xt .

a) XM--:>suPt
(a) comes from 6-44. Note that zq,x is the same

as in the proof of 6-44 for q~2, while Zl,x=Xx and ZO,x

Y~ t(q) is a part of the components of z~+~'x if q~l, ~f Z~+1,x if q=O. is just a part of Z1,x Now,

So

in the proof of 6-44 we saw that zq,x satis-

fies Equation 6-2 with (A'-(r-q». So by 6-16, zq,xEH x whenever r-q~2. Moreover if r-q~l, 6-2 as well with, at least,

t

t

then zq,x satisfies

(A'-l). Then the estimate

5-11 gives D(zq,xtu

< c (l+lxl),

T LP

P

since the initial condition Zci'x is (x,I,O,O). Hence we have (b) and

(c).

Then the results of Section 4 immediately yield: 6-48

THEOREM: Assume (A'-jJ and 6-38, and set x

det(DX t ) and (=00

if P(Q~=O»O).

(6-49)

a) If j~3 and Q~#O a.s., X~ admits a density y~Pt(x,y).

b) Moreover, pt(x,.) is of aZass c r , provided: - either j~r+d+3 and q~(2r+2d+2+E)<00 for some E>O, - or j~r+3 and q~(2d(r+l)+E)O. a) Moreover, ded:

(x,y)~ Pt(x,y) is of alass C r ,

provi-

CALCULUS OF VARIAnONS

either j>r+2d+3 and bounded for some E>O. - or j~r+3 and for some E>O.

x~

x

x~qt(2r+4d+2+E)

79

is locally

x

qt(4d(r+1)+E) is locally bounded

d) Moreover: (i) If j~2r+4d+6~ if for every bounded subset Acm d sup >t t

x

O,x

EA qt(4r+8d+8+E)O, and ¥v E [

for some constant ~>O, then class C r on (to' T] xlR d xlR d . (ii) If

j~2r+4,

0,1)

(6-50)

(t,x'Y)~Pt(x,y)

is of

if for every bounded subset Acm

d

,

sUPt>t ,xEA q~(4(r+1)(2d+1)+E) <= for some E>O, and if c=O (~.e., the Poisson measure in 6-2 does not intervene),then (t,x,Y)~Pt(x,y) is of class c r on d d (to,T]xlR xlR •

Proof.

(b),

(a),

(c) immediately follow from Theorems

4-7,4-19,4-21, and Lemma 6-47.

(d) also follows from

this lemma and from Theorem 4-31, provided we prove the next result. 6-51

LEMMA: Properties 6-50 and (A'-j) for some

j~2

imply that the family {~~=x~} meets 4-26 for j, and if c=O we have ~ u (x)=x in 4-26. Proof. Recall first that the extended generator of the Markov process XX (see [ 13]) operates on C2 (Rd) as p

1

T

2

Lf(x) = a(x)Dxf(x) + Zbb (x) Dx2 f(x) +

J[ f(x+c(x,z»-f(x)-Dxf(x)c(x,z)] dz E

(6-52)

80

MALLIA YIN CALCULUS

t f(Xx)-f(x)-J Lf(Xx)ds

and

0

t

under (A'-l),

s

is a local martingale.

IXxr;'En <",L P , while

for suitable 1;,6. Hence

lLf(XX)

Since

f

and the abo-

and

t

E(f(X~») = f(x) +

J

o

is continuous,

and the abo v e y i e 1 d 5

6

ILf(x)I~1;(l+lxl )

I~TEn p <'"L P ,

ve local martingale is a martingale,

Now,

E(Lf(Xx)ds. 5

s~E(Lf(Xx» 5

t hat t ......> E ( f (X ~»

is continuous, is d iff ere n t i a -

ble, with derivative E(Lf(X~». So it remains to prove that m,A i ,$

with

f

is of the form 4-28,

satisfying (i)-(iii) of 4-26. Second or-

u

der Taylor's formula with integral rest yields f(x+c(x,z))-f(x)-Dxf(x)c(x,z)

=

1

J

?

(l-v)
° 6-52

so that

dv,

x

is exactly 4-28, provided we put (n being

the auxiliary function in (A'-j»: U

{O } U ( E x [ 0, 1 ] ) ,

2

m

n (z)G(dz)x(l-v)l[O,l] (v)dv, a,

A2 u

=

.!.bb T 2

if u=O,

0,

and

!jIu(x) = x+vc(x,z)

i f u=(z,v)

EX[O,l].

The regularity assumptions in 4-26 are then clearly met by these terms L

2

(recall that the function n is in

(E,G». Finally,

the last claim is evident, with the above

definition of !jIu'

Section 7: PROOF OF THE MAIN THEOREMS VIA BISMUT'S APPROACH

§7-a.

INTRODUCTORY REMARKS

Our aim now is to deduce Theorems 2-14, 2-27, 2-28, 2-29 from Theorem 6-48. However,

the settings in Sec-

tion 2 and in Section 6 are different, and we first have to conciliate them both. This is presently achieved,

through the following remarks.

1) The driving terms are an m-dimensional Brownian motion Wand A+1 Poisson measures Pl, .•. ,P A and P, as in 2-1. It is no restriction for the results of Section 2 to assume that the filtered space supporting W,pa,p is the aanonical spaae, as defined in §6-a, but with A+l Poisson random measures instead of one. 2) As mentionned in Remark 2-16, ken Assumptions (A-r)

it is possible to wea-

into (A'-r), for the coefficients

of Equation 2-2. However,

the space E on which the last

Poisson measure P sits is an "abstract space", and

z

there is no differentiability in ent

for the coeffici-

c. Hence, one should state Assumption (A'-r) as

follows: 7-1

ASSUMPTION (A'-r): a,b, and each c a ' meet 6-3 (the

latter with some TJ En a

c

2..3'

<'"

LP(E

E».

a ' =a

The coefJ~icient

meets (i) and (iii) of 6-3 with some TJEn2
~

(A'-r).

81

MALLIA YIN CALCULUS

82

3) All results of Section 5 are of course valid with A+l

poisson measures instead of one.

4) Now we describe the changes to be undergone in Section 6. We will perturb each measure ~

a

>

but not

~.

we start with a function P :

So for each a=1 •... ,A E

~

a

[0,=) meeting 6-9. and with a function va meeting

a 6-10 relatively to P a . Then Equation 6-15 reads

The reader may also think that one "perturbs" but with the function v=O!

)J

as well.

so all of Section 6 goes

through without modification, except that every integral with respect to

~

(resp. 0)

should be replaced by ~a

a sum of integrals with respect to

and

~

(resp.

Pa

and P), with the convention v=O. In particular, VXX is still given by 6-32, with 6-31 replaced by KX = D a(Xx)~t + D b(Xx)wW x

-

x-

+ ID c (Xx)*O a x a

-

a

(7 -2) + D c(Xx)*P.

x

-

6-34 also holds, with (instead of 6-41):

with g

satisfying 6-38-(ii) relatively to p . The funa a damental formulae 6-35, 6-36. 6-37 remain unchanged, as well of course as Theorems 6-44 and 6-48.

§7-b. EXISTENCE OF THE DENSITY We begin with auxiliary results.

BISMUT'S APPROACH

7-4

LEMMA: Under (A'-l).

83

the following (where B,C

are

C1

defined in 2-10)

defines a process with values in the set of all symmetric nonnegative dxd matrices, and this process is nondecreasing for the strong order in this set. Proof. Since Band Ca are nonnegative symmetric matrices, and PC1~O,

it is clearly enough to prove that

R~

is finite-valued for all t
B(X~)*t is finite-valued.

In view of 2-10 and of (A'-i)

again, ICa(x,z)I.5'(I+lxl

e' )(1+n~(z» g a (x, z)

-2

for some constants ~',e' and nti En 2 g =ldet(I+D c a xa [(1+lx

x

-

1

)1.

(7 -6) I {g ~


a

(x, z) ;! 0 }

LP(E ,G ), and a

a

Since pEL (E ,G), we have a.s. a a a

I 6' )(l+n')p a a*II]T a

Hence, due to 7-6,

< "".

it remains to show that for almost

all w, x

ga(Xt_(w),z) ~ yew) ,lIa(w;dt,dz)-a.s.

(7-7)

on the set {(t,z): ga(X~_(w),z);!O}, where y(w»O.

Let gt(w)=det(I+~K~(W».

In view of 7-2,

outside a P-null set, 7-7 is equivalent to saying that for all t'::T,

either Igt(w) I~Y(w), or gt(w)=O.

(7-8)

84

MALLIA YIN CALCULUS

x x x x,. Tn
x x Tn (w)
for

Thus 7-8 is met with y(w)=

Moreover gt=O if t=TX. n inf 0<.N() Y 0 (w), where III X

N(w)

-

TN(w) (w)..2T
is such that

since N(w)
and the proof is finished. LEMMA: Assume

7-9 p

a

>0 on E

(A'-l) and (B)

(see 2-13)J and that

for all a=l, ... ,A. Then P-a.s. the matrices

a

R~-R: are invertible for aZl 0..2s
For simplicity, we omit the superscript

tI

X

1t

•

Let s s,Y (w)B(Xt_(w»Y(w) ~n}' { (w, t , z) : t> s , (X t

(w), z) E r ' ,

T a l (w)Ca(Xt_(w),z)Y(w) > n}'

-

Y

v = lim

=

.zn

Z

=

1

n

Vn

+VO, *t +

IV~t +

Ip

V

Ip

a a

a a

1

lim

a

Vll

~~

o

+Vn a

a'

IV *~ . a

a

As p ELl(E G) zn and a a' a ' sing processes, and Zn t over 7-5 yields for t>s:

Z

are finite-valued increa-

increases to Zt as n+ oo . More-

BISM1JT'S APPROACH

yTR Y _ yTR Y > (yTB(X )Yl s

t

Vn

~t)t

+ I(yTC

a

1.

>

-

n

85

a

(X )YP 1 -

a Vn a

*~)t a

zn,

(7-10)

t

Let 8>0 and u 8 =e- 8Z . Then Ito's formula for processes with finite variation yields 6 U = 1 -

8 8U_ 1 V*t -

\' 8 L8PaU_1V ~~a a -epCla

+ 'U 8 (e L a 8

6

= 1 - e~_1v*t - IU_(I-e

-1+8p)1 "'lJ a V Ct -8P a a

)1 V

a

a

"'U a

(7-11)

'

-8p <1 and 0 <1-e a <ep , which is G - integrable. a a So when taking expectationsin 7-11 we may replace ~a by

Now 0
8

va (noting that Va is predictable), We obtain for t>s:

E(U~) = 1 -

t

J

E{8U~lv(r)

s

+ LU:

J

a

Ea

t

< 1 -

J s

-8p

(z)

a

(1-e

(r,z)dz}dr

)IV a

E{81 v (r)1 {Z =O} r

,

J

-6p

(z)

a

+ Ll{Z =o} Cl-e ex r Ea

1 )1V (r,z)dz.dr. Cl

Each individual term above is increasing in 6, so we may pass to the limit as OX<x>=O, we get

8+~.

With the convention

(because P a >0) :

P(Zt=O) ~ 1 -

t

J s

E{+~ I V (r)l{Z =O}

(7-12)

r

+ Ll{Z =O} flv (r,z)dz}dr. a r Eex a a In the following, we use the notation Wax , Nx , Nxz of 2-11 and 2-13. We have V={(w,t):t>s,Y(w)¢N X (w)} (recall that V=

~V

n

), so if

t-

86

MALLlA YIN CALCULUS

v ' : = { (w, t) : t > s , Y ( w) ft-W ~ Ct

nx (s, T)

VU(UV') = a a

t_

()}, Ass u mp t ion (B) y i e 1 d s W

.

0-13)

SinCe V =UV n we also have a n Ct' a = {( w , t , z) : t> s , z E r' X ( ) ' Y ( w ) ~N x () a, t- W t- W

va

Therefore if t>s and Y(w)~WXat_(w) we have

,

z

}.

r'a,Xt_(
{z:(w,t,z)EV}, and thus (recalling 2-13): 0.

J

IV (w,t,z)dz ~"'XIV,(w,t) 0. a 0. a for all w,t such that Y(w)ft-WXt_(w),t>s. E

Hence 7-13

yields for r>s: '" IV (w, r) +

If

IV (w,s,z)dz

aEo.

a

~ ",I V (w,r)

+ "'x(II v ,(w,r» a a

(note that we have not used the (usually wrong) fact that V'EFXB([O,T)).

==

a

t>s: P(Zt=O) ~ 1 -

f s

Substituting in 7-12, we get for

t

E("'l{Z =O})dr. r

This is only possible if P(Zr=O)=O for almost all

r

in (s,t). Since Z is increasing, we deduce that Z>O a.s. on (s,T]. Finally, consider 7-10 and recall that Zn tZ . If t>s we have Zt>O a.s.,

T

T

n

so Zt>O for

Y RtY>Y RsY a.s. Then,

n

large enough,

so

for almost all w, yew) does not

belong to the kernel Lt(w) of the linear map Rt(w)-Rs(w).

Because of 7-4, Lt(w)

inc~eases

as t de-

creases, and we set Ls+(w)=Ut>sLt(W). Then we have that Y(w)ft-L

is F -measuras+ =s ble (in an obvious sense) and since the above property 5+

(w)

for almost all w.

Since L

BISMUT'S APPROACH

87

holds for each unit random vector rab1e, we deduce that L

Y that is F -measu=s (w)={O} for almost all w: this

s+ implies that Rt-R s is a.s. invertible for all t>s. Finally, let nO be the set where Rt(w)-Rs(w) is in-

vertible for all s
-

r

r

is invertible. Because

R is increasing for the strong order (see 7-4), Rt(w)-Rs(w)

is a-fortiori invertible, and the claim is

proved. For the next result, we recall that KX is defined in 7-2, and that T~=inf(t:det(I+AK~)=O)

(see 6-35).

LEMMA: Assume (A'-J) and (B).

Then if tE(D,T], the law of x~ under the restri~tion of the probability to the set {t
A

is a Borel subset of lR d

with Lebesgue measure 0, (7-15)

Proof. We will apply the following extension of Theorem 6-48-a:

in view of 6-45 and 6-47 and of the first part

of Theorem 4-7, and since (A'-3) holds, the claim will be proved if we can choose u a)

6-38 holds,

b)

the matrix

and

va

so that

and

DX:

is a.s.

invertible on the set {t
For proving (b) we will use Lemma 7-9, which requires Pa>O on Ea' Since Ea is a countable union of Sa-diit is easy to find a function P a which meets both 6-9 and Pa>D on E . a amounts to choosing funThe choice of u and

mensional rectangles,

ctions

f

and

ga

which meet 6-38, and it intervenes

88

MALLIA YIN CALCULUS

in the expression 7- 3 for the process HX. Now, 6-37 yields

while V ~ is invertible for t
fo

("1X x

s-

)-I(I+6K x )-l dH x is a.s. s s

(b) amounts to

invertible 0-16)

x

on {t
and

f

choose the following functions f and and y E m d 0lR d and zEE a ):

ga

ga

we could (here, xEm d

f(x.y) = b(x)Ty-I,T

(=0 if Y is not inver-

-

T

g

a.

(x.y.z) = D c (x.z) za. (=0

(I+D c xa

if Y or I+Dxca are not invertible).

Then the process in 7-16 becomes,

f

t

o

t ible) _IT_IT O - I 7) (x,z)) 'y ,

("1';< )-1 dR x

s-

s

("1..;x )-I,T

in view of 7-5:

for

s-

which in virtue of Lemma 7-9 is clearly invertible a.s. Now,

the functions 7-17 do not meet 6-38. To over-

come this little difficulty, we proceed as follows. We define the following functions:

md 0lR m

h(y)=(I+iyi 2 )-1:

h'(y)=(1+iyi 2 )-1 a. h": m d 0lR d .... [0,1] and only if y""'-> y

-1

T

y

....

md 0lR Ba

(0.1] .... (0,1]

is such that h"(y)=O if

0-18)

is not invertible, and ",,'

d

, h" (y) is of class Cb on lR ®lR

d

(there are plenty of such choices for h"). Then f and

BJSMUT'S APPROACH

89

ga being given again by 7-17, we set:

= f(x,y)h(b(x))h"(y)

f(x,y)

ga(X'Y'z)

= g a (x,y,z)h'(D a z c a (x,z)

}

(7-19)

h"(I+D c (x z)h"(y). x a ' It is then easy to check that Moreover, t

f

and

meet 6-38.

the process in 7-16 reads as

or (17X s- )-1 x

d~x (I7X x )-l,T s s-

x for t
(7-20)

where

Since h>O, h'>O, and h"(y»O for a

AX

y

invertible,

Ax

comparing 7-21 and 7-5 yields that Rt-R s is invertible x x whenever Rt-R s is 50, and the latter is true a.s. when t>s by Lemma 7-9. It is then obvious to deduce (using 7-20)

that 7-16 holds, and we are finished.

7-22 Proof of Theorem 2-14. We will deduce the existenx

ce of a density for X t (where tE(O,T]) from Lemma 7-14, via a Markov process argument which has already been used in this context by Leandre [18]. It is well known that the solution to 2-2 is strong Markov; however, in order to properly apply the strong Markov property at time T~ we need to be somewhat careful. Firstly, we can indeed assume that the time interval is

lR+ instead of [O,T];

this slightly simplifies

the description of the shift semi-group, which goes as follows:

6t

is the unique map from the canonical space

n into itself, such that Wt+s-Ws=Wso6t and

90

MALLIA VIN CALCULUS

~(w;(t+ds).dz)=~(etWtds,dz),

and similarly for all ~a'

The independence and stationarity of the increments of w'~a'~

-1

then yield that poe g =P for every finite stop-

ping time

S.

Secondly,

let

S

be a finite stopping time;

then

woeS is again a Brownian motion, and we have

J

S+t

S

t b(Xx)dW = b(XxS + )dW o9 S s sOu u

f

up to a null set.

Similar equalities hold for the other

terms appearing in Equation 2-2, and thus

up to a null set. So the uniqueness of the solution to

2-2 yields x

a.s. for all t, where Y=X S '

(7-23)

The same argument applied to 7-2 also gives

Next, we will apply this to the stopping time S=T X defined in 6-~, and we put again n

x

Y=X s ' Then

6-35 and 7-24 yield Tx n+1

=

T X + TY 1 0 9 T~ n -1

x< } a. s. on {T nco.

-1

(7-25)

We have seen that P o 9 S =P, and 9 S (~) is independent from ~S on {S }. Then. if A is a Borel subset of ]Rd 7-23 and 7-25 imply

BISMUT'S APPROACH

f P(dw)1{TX()

n w
If moreover

A

}P(X

91

Y(w) Y(w) x EA,T l >t-T (w». t_Tx(w) n n

has Lebesgue measure 0, we deduce from

7-15 that (7-26) x

x

Finally, P(Tn=t)=O for all n, because Tn is one of the jump times of one of the Poisson measures summing up 7-26 on

n

x~

tly saying that

~a

or

~.

So

x

gives P(XtEA)=O, which is exachas a density.

§7-c. SMOOTHNESS OF THE DENSITY Again, we begin with auxiliary results. 7-27 LEMMA: Under (A'-2) and (se) p<'"

the function

(see 2-26)~

for alZ is Zocally boun-

x-E(I (v~)-ll;p)

ded. Proof. Recalling that + D a(xx)V:?*t + D b(Xx)VXxl\W X

-

x

-

-

-

+ IDXCCt(X~)V~'tPCt+ DxC(X~)V~ .. P, a

it is well known that yx=(V~)-1 is the solution to

where, due to (SC), we can define Ax = _ D a t X

(rt- )

+

!

Db' i ( Db' i) T (Xx ) i=l x x t-

+ f[D c(I+D c)-lD c](Xx E

+

x

x

x

t-

,z)G(dz)

~f [D c (I+D c )-1 DxC ] (Xxt ,z)dz

LE

a a

x a

x a

a

-

92

MALLIA YIN CALCULUS

EX t

=

-D b(Xx )

t-'

X

-D c x

a

x (X t x

- D c (X t x -

-

,z)[I+D c x

a.

(X

x-I ,z)] , t-

x-I ,z)[ I + D c ( X t ,z ) I • x-

(to verify this claim, it suffices to define

yX

as

the solution to 7-28, and to apply Ito's formula to the product

yX=VXXyx:

one obtains that yX is the solution

to a linear equation,

to which the constant process

is also trivially a solution:

I

hence YX=I). d

Now, Equation 7-28 is of type 5-3. If lR JR d1x •.. xlRdq is the grading in ( A , -2), it is easy to recognize that the coefficients of 7-28 are graded, d d according to the grading of JR ~lR described in [/3] of the proof of 6-44, and due to (A'-2) and (SC) x "'X tisfy 5-9, with Zt and Zt as follows:

they sa-

0, X*

for some constants 1;,9>0. Since D (X )TII is locally LP bounded in X for every p<~, it is easily deduced from the estimate 5-11 that x,..." n (y x* )TD

is locally bounded LP

as well.

LEMMA: For every symmetric positive dXd matrix A and every pE(O,m): 7-29

r(p)d

e

<

det(A)P (where

f

-xTAx dx < f(p)d d d(2 P- 1)12. det(A)P

is the usual gamma function).

Proof. There is an orthogonal transformation of lR d to itself that changes neither but transforms may suppose that

A

in-

T

det(A),lxl. nor x Ax,

into a diagonal matrix. Hence, we A

is diagonal, with entries AI>O, •. ,

BlSMlJT'S APPROACH

93

Ad>O. A simple substitution gives

rep) Hence d II i=1

r(p)d det(A)P

2 d +"" -Ai(xi) II dx. Ix.1 2p - 1 e ~ ~ i= 1 -""

rJ.£l

f

),P i d -xTAx ( II Ixil)2p-l e dx i= 1

JlR d

d

r d (2p-l)( II cr )2p-1 i= 1 i

!""r d - 1 dr

o

J

00

<

r

2 d 1 p

-

dr

o

J

~

_r2~TAo v

do

(7-30)

S d-l

(since 101=1 if oES d _ 1 , the d-dimensional unit sphere, so

IOiI21). Finally,

f

d

I x l d (2 p -1)

7-30 equals T

e- x Ax dx,

lR

and the first inequality is proved. verse, we notice first

To prove the con-

that the function f(o)=

II1
7-31

LEMMA: Assume (A'-3) and (se) and

2-24) for some ;,8>0. and "let pc£. (SB-(r,.8)); set

E.

(SB-(r,~8))

0 be like in

(see

?vL~LLlAVrl'\

'14

k(

,X

•• _,

;-;,<x) ('lXx)-l ' • _ ' _

""{x + r,.,(Xx ) r.. • I 'i. .... 0,

where

\l

-

-

,

Z

k:(x,y,z)

'"

x -np

x~E(det(S~)

Proof.

Since

boundec

\". :J.. . _~ - , Z

)

-

C:~e

Ec~'eZ-

a

(see 7-27),

. ('.{

C). Z *'.~

Ci

an.:i satis.fy

Iv

I ~" ~

> .-' '. _ _ .--:--;.-'-----;;-;;5 ")(1+ lyI 5")'

0-33)

- (1+1:-::

is locally bounded.

are bounded and

k, k'

(7-32)

)(",,,x)-l,T ~~. :J

(0, 1 J and k': c!

X

k (x, y) ,

(VX

M

-

(JRd 01R d)x:: -+ (0,1] . a ecn..'3tcnts 6',01'>0: JRd

B(yx)('7,(x)-l,T " _ \ •• _ ;I- t

)(",(X)-1 "4.

k:1R dx (lR d ®1R d )-+

CALCCLUS

(7 -3 4 )

x -1

is locally

(9X_)

if we compare 7-32 and 7-5 we ob-

serve that SX is a process taking values in the set of dxd symmetric nonnegative matrices and is non-decreasing for the strong order in this set. We now break the Step 1. We have For x,yElR

d

into several steps.

proo~

(SB-(t,e»

for some broad functions f

Then 2-25 becomes,

Elyj2

1

for all x,y we can delete this value

from I without altering 7-35.

If

~

_1

pair (x,y), . all zEE CL

9' (1+'1

a

•

(7-35)

0'

l+lx'

aEI ~a(x.y)=O

.

if I={1,2, .. ,A}:

E;~l(X,y):..

I

yTB(X)Y +

If

a

set

:x

a

(x,y»O for some T

then f a (z)
)2 where

5'>(l and

r

. c:

En

2..::p

<<<>

LPCE

:x'

G) Cl

'

while

BISrvfllTS APPROACH

Thu5 clearly f.E:L1(E,G),

"Eo,-' LP(E G). u 1 ~p ~ 00 .")' a tic u 1 a r G ( f > 1 ) <".

(~,9)-broad.

is still

~a(x,y)

increases

and

Moreover,

i~plies

replacing

f

that '

a-

if

:).

w" nay replace f~ by


all

.:1

f:;'(I',,:;

(;,S)-broadne~s

In other words, we ffiay and will assume that

.

is

u

without altering neith"r 7-:5 nor the of f

That

Al

Y •

fa(z)dz

and

':t

~l

:X'

Set

<1.

Ea I'e have Y(X~'r,

~ar-

f

bv f

0.

does not alter 7-35.

SO

to say, we may assume

!

2-3~

lerrma

S

In

Cl.:t

t..!

Th u

O.

(!

95

~or

CLEI:

O
r

•

f

(l

(z)dz

Y

£

\

< 1; ' - -

(7-31',)

\ A+ 1 •

Eo. S teD tl

1:n til fur the r

2.

not ice,

Let oES d _ 1 be given,

x ".

and

,,"' e d r

0

p

the sup e r s c rip

t

set

v; >

>yf (z) a

a

;

t-

'

! VX -1 ' T (w)o ·2 1(1+ !' X t-

if aEI, and V; ; V

y 2! YX -IT '(w)c I')-/(l+'X t

0

t-

(w) I -: )

-

(LC)

!o )f,

for all zEE

:x

}

i f CL1'I,

V' \ ( V UV ' U. • • UV ' ) . a 1 ·::;-1

t, ~ Since "<1. 1• \ l+A (V,(V)

a

-1)

LtC

we deduce frOID

i-35

that

is a predictable partition of (7-37)

of Qx[ 0, T]

96

MALLIA VIN CALCULUS

Since all the integrands of 7-32 are nonnegative

mat-

rices, we deduce from 7-33 that

crTSo

>

H crT~x=lB(X_)vX=l,Tcr I V. t +

\'

L

aEI

T

-1

-1

H cr ~X_ Ca(X_)VX_ '

T

cr o a 1V ~~a a

>

1+lx f

U~X!_" the operator norm of

Call

l~x-l,T(J12 t-

>

linT 11- 2 t-

(because

9X!_:

a IV a

*

j.J

a

•

then

101=1). Thus i f

o,-l(l+lx 10)(1+lx 1 0 ") s s (7-38) 0 ", o+ln 15 ")191 s s

sUPs
1-

then

the stochastic integral Za=f IV ~p a a a

Since f El1(E ,G ) a a a exists and Za =

(f

a

1 V ) it"IJ

a

a

(recall 7-36). Thus with Z=LZ a , and using 7-37, we get a (7 -39)

Step 4. We shall now compute E(Y:), $>0.

Ito's formula gives

where Y·=e- tZ and

(as in 7-11);

BISMUTS APPROACH

97

define the nonnegative numbers: -~f

{l-e

(z)

)dz

(l

(7 -4 0)

and rewrite IP -$fa. (7 -41 ) {Y (e -1) Iv *0 a a -41 f a Using again O
L

aEI

Equation 7-41

IY~I~TEn

in order to obtain that

q <'"

Lq,

and

since f En 1 Lq(E ,G ), Lemma 5-1 yields that the loa
I

+

aEI

t

n~E(J Y¢IV (s)ds) a

a

s

0

t

I +

(max~EI nIP)J E(y
a

°

s

and Gronwall's Lemma gives ~

(7 -4 2)

< exp ( t max exE I flex)· Step 5. For crESd_1, u>O, v>O, h

u, v

(cr,t) = /"sU-I 0

Using 7-39,

t>O,

set

ds {E[exp -Sl'tCcrTStcr»)}v. (7-43)

7-40 and 7-42, we obtain

E{exp-~Ft(crTStcr)}

< exp(-ty¢ +tmax aE1 <

max exp[ -t J Cl-e aEI Ea

n!)

-¢ia(z) ) dz)

Taking $=ys above, we deduce (with notation 2-21): h

u,v

(cr,t)

<

J'" s u-l o

< y

-u

I

-ysf ds max exp[ -vt j(l-e aEI Ea

aEI

y (u,vt,f ). a a

(z)

a

)dz] (7-44)

98

MALLIA YIN CALCULUS

Step 6. Let p>O, nElN*.

7-29 gives

det{F S )-n p < t

t

S inc e. n E(

II

n

~

!W i !)

i= 1

/

II

i

e

1

E{!w i !n)l n

for all r.v. Wi' we obtain from 7-43 and 7-44 by taking expectations: 1 J~ J~ pd-1 d •. (5 1 •• 5 ) ds 1 · .ds n 2 r(p)n 0 0 T n n -ns.F aiS a. 1/ n II E(e ~ t t~) n dal .. do 1=1 n

2nr~p)ndnn~d {~d-l h pd ,l/n(cr,t) <0 {I (dtf)}n p,n,y,d aEI Ya p '-0' a <

do}n (7 -4 5)

where 0p,n,y,d is a constant depending only on p,n,Y,d. Step 7. Now we make the dependence upon the starting point

x

explicit. Let p,t,n with pd<~, ~>e, and nE

n-

(p+~)d<~. Since each f is n a {~,e)-broad, 7-45 gives a constant 0 (not depending on x) su.ch that for all xElR d , and if p '=p+~, IN*

• Let 00 be such that

E(det(F Xt Sx)-np') t

X

-

< 6.

11-

X

We know that xlr-» I (X )T" x -1 *" x fr-~ I ( (V X) ) TU

(7 -46) ~

and x/orll (V}!: )T1 Lq

and Lq

( by 7 - 2 7) are 10 calI y b 0 u n d e d for Lq x all q<"', and so is xt-'>BFtl because of 7-38. We also x -1 x d Lq x-I have deteSt) = (F t ) det(F~St) ,hence 7-34 follows

BISMUT'S APPROACH

99

from 7-46 and the fact that p'>p.

7-47 Proof of Theorems 2-27, 2-28, 2-29 under (A'-r). We will apply Theorem 6-48. We assume (SB-(t,e»

with

some P a like in 2-23, and

(SC) and at least (A'-3), x according to 6-49 we put qt(i)=E( I det(DX xt ) I- i ). The choice of firstly consider

u

h.h~,h"

tant showing in (SC), se

h"

and

and

is as in Lemma 7-14. We a as in 7-18. If I;; is the consv

it is obviously possible to choo-

so that

={ ~et(y)2

h"(y)

1+l yl 4d

if

Idet(y)1

if

Idet(y) I

<

(7 -4 8)

r

..2-

2•

Then f,ga are given by 7-19. Because of 7-2 and we have

(SC),

Idet(I+~Kx)I~1;; identically, so T~=OO in 6-35 and

6-37 yields DX x t

=

t

VX X t

J0

(VX x ) -1 s-

(I+~Kx) dHx. s

s

Then due to 7-20, we get t

J

DX x = VXX (VX x )-1 dR x (V~ )-l,T t t 0 ss s' where

ix

is given by 7-21.

Now,

if we compare 7-21 and

7-32, we get (7-49) prov ided k(x,y)

= h(b(x»h"(y)

ka'(x,y,z) = h'(D c (x,z»h"(I+D c (x,z»h"(y). a ZCl XCl In virtue of 7-18 and 7-48 and (A'-I) and

(Se), one

easily checks that 7-33 is met for some constants 6',

100

MALLIA YIN CALCULUS

0">0. Hence by Lemma 7-31, t

-n~e,

pd<s and

if nElN*

and pE(O,o:»

there is a constant 0A d

each bounded subset

A

of 1R

,t

for each t>O and

with

for all xEA,s~t

E(det(S:)-n p ) 20A,t

satisfy

(7-50)

s ~ det(Sx) increases). By 7-27, s x,.,..." sup T E( det(17X x ) I-q) is locally bounded for all s< s q. Mor;over det(DX x ) = det(17X x )det(Sx) by 7-49. (recall that

I

s

s

s

Therefore 7-50 yields that

provided p'
and where 0A' p'

x",,-,>suPs>t q xs (i)

, t

is a constant depending

In other words, is locally bounded if } (7-51)

and p'de nObserve that the conditions in 7-51 are equivalent to saying that t~e and Z;>d i

I[tJ.

Hence parts (b),

(c),

(d) of Theorem 6-48 give Theorems 2-27, 2-28, 2-29 respectively

(in the setting of 2-29,

se be replaced by 2-30).

6-50 should of cour-

CHAPTER IV MALLIAVIN'S APPROACH Section 8: MALLIAVIN OPERATORS This last chapter expounds

}~lliavin's

(or Stroock's,

or Shigekawa's) approach for finding a (smooth) density x for the random variables X t in 1-4. Sections 8 and 9, which are independent of what precedes (except for the notation of §6-a and the definition 4-4 of an integration-by-parts setting) are concerned with general facts. Section 10 applies to stochastic differential equations. The (very short) Section 11 is a replica of Section 7, in which the main results of Section 2 are re-proved. Finally, Section 12 contains some heuristic remarks about Malliavin operators and their connections with Bismut's approach. As a matter of fact,

the present Section 8 is con-

cerned with the same abstract setting as Section 4:

it

explains how Malliavin's ideas can be set for constructing an integration-by-parts setting which works simultaneously for a "large" collection of random variables. Stroock also has introduced "abstract" Malliavin operators in [26], following a slightly different approach.

§8-a. DEFINITION OF MALLIAVIN OPERATORS Throughout we fix a probability space (Q,~,P). Recall 2 n that C (R ) denotes the set of all functions on lR n p of class c 2 , whose derivatives have at most polynomial

101

MALLIA VIN CALCULUS

102

growth. DEFINITION: A Malliavin operator (L,R) is a linear

8-1

operator

L

on a domain Rcn

p <00

LP ' taking values in

and such that: (i) R is stable under C 2 : i.e., p

then F(tfl)ER. (ii)

~

is the a-field generated by all functions

in R. (i i i)

is self-adjoint

L

i. e.

E(H'I')=E('l'L¢)

for all tfl,'I'ER. (iv) L(tfl2) ~ 2tflLtfl for tflER;

this amounts to saying

that the symmetric bilinear operator r associated with L

by r(tfl,'I') = L(tfl'l') -

'l'L~

tflL'I' -

(8 -2)

on RxR is nonnegative. 1

n

(v) I f tfl= (tfl , •.• ,4»ER

n

?

n

and FEe; (R ), then

n

L (Fo 4»

L _a_F(~) 14>i i= 1 () xi 1 n (l2 i j + 2 L (lx.Clx. F (tfl)r(tfl,4». i,j=l ~ J

(8-3 )

Here are some easy consequences of this definition. Firstly, one notices that 8-2 and 8-3 are compatible. Secondly, an easy computation based upon 8-2 and 8-3 1

k

shows'that if 4>=(tfl , ... ,4»eR k

L

a

-a-F(4))

i=l We also have r(tfl,'I')En

k

2 p

k

, 'I'ER, FEe (R), then

r(~

i

,'1').

(8-4)

xi p <=

L P if 4>,'l'ER. From (v), L1=0;

hence (iii) and (iv) yield for 4>,'I'ER:

~lALLIA YIN

E (U)

=

OPERATORS

103

0

}

(8-5)

The nonnegativeness of r implies

(8-6) which in turn yields

(8-7) We also deduce from 8-5 that (8-8) The following fundamental lemma is well known for self-adjoint operators:

LEMMA: Let {~ n }cR with ~ n ~ 0 and L~ n ~ 6 in L2. --Then 9=0 a.s' 3 and r(~n'~n) + 0 in Ll. 8-9

Proof. For all

~ER

we have

E(~LWn)=E(~nL~);

E(~9)=0,

the limit as ntoo yields

going to

and 8-1-(li)

E(r(~n'~n»=-2E(~nL~n)'

9=0 a.s. We also have

implies from

which we deduce the second claim. Finally,

the Malliavin operator (L,R) pertains to

the construction 4-4 as such: PROPOSITION: If ~=(~l, ... ,~k)ERk. the foZlowing

8-10

set (o~,y~,H~,y~) is an integration-by-parts setting for ~:

a;j H

~

r(~i,~j),

yj

W

=

-2U j

=R, Ilj(~) = - r ( ~ j , ~)

for 'I'ER.

}

(8-11)

Proof. These terms clearly meet conditions (i)-(iv) of

104 4-4

MALLIA VIN CALCULUS (4-5 follows from 8-4). Let fEC 2 (Rk). p

Then 4-6 fo-

llows from k

a!

L

E('!'

i=l E[

f(~)cr!j)

(by 8-4)

E['I'r(fo,.pj)]

i

'i'L(~jf(
E[q)f(
-

HjL(fd)

f(¢)L('f¢j)

-

'!'f(
(by 8-2) (by 8-5) (by 8-Z)

§8-b. EXTENSION OF MALLIAVIN OPERATORS Despite (i) and

(ii)

of 8-1,

the a-priori domain R

might be too small to contain the variables of interest. This is why we now proceed to extending

R,

essentially

following S troock [26] . 8-12

DEFINITION: We call H2 the set of all

~EL

2

for

which there is a sequence {

n} is Cauchy in L ; we then denote by L L 2 -limit of

Ln

(due to 8-9,

pend on the particular sequence

II

2

un z

+ DLH

L

(the graph-norm of

8-13

PROPOSITION:a)

such the

this limit does not de{

For EH Z we set

2 L

(L,H 2 )). (H 2 ,U .U~2) is a Banach 8pace~

is the cZosure of R for the norm D.OH

and

. 2

b) r can be extended uniqueZy as a ·continuous biZinear symmetric operator:H 2 xH Z ~ L1. Moreover~ this extension is nonnegative.

MALLIAVIN OPERATORS c)

(L,H 2 ) is a self-adjoint

105

(closed) operator on

L2, and 8-5, 8-6, 8-7, 8-8 hold for all ~,~EH2_ Proof. {~

n

(a)

is trivial. To prove (b)

lcR going to ~ and! for E(lr(~

n

, ! )-f(
n

m

,~ )

m

< E(lr(1

<

2{nl +

n

-~

20L4> -LI n

(c),

let {

I) n

-4>,'!' )1)

mn

U

m L

20'1'

H

n L

+ E(lr(4>

2"L'I'

0

n L

,'1' -! )\) mnm

211/2

2{nl " 2"t. U 2"'1' -'I' 0 20L'I' -L'I' 0 2}1/2, m L

which goes to limit,

U

m L

and

m L

0

as n,m+=.

say f(.,~),

converge to I

n

m L

n

m L

Rence r(4)n''!'n)

goes to a

in L1; moreover if {e'l,{'I"}cR also n

and'!' for n.D~2'

n

the same argument appli-

ed to [(4)

,'1' )-f(I','I") shows that f(I','!") + nl,'!') as n n n n n n well. Hence r is a bilinear symmetric nonnegative opel

rator:HZXH Z + L , which extends 8-2. Therefore we have

8-6 and 8-7, and also 8-5 and 8-8 by passing to the limit. Finally if {4> l,{'!' }CR Z converge to I n

n .UH'Z'

r(
n

and ~ for

again the same argument shows that r(1 ,'!' ) ~ 1 n n in L • So we have (b), and (c) readily follows

from all what precedes. Of course the domain HZ is not in n LP, and it is Z p <00 not stable under c p ' so (L,H Z ) is not a Malliavin operator. So we presently restrict the domain of (L,H Z )' Firstly,

set

(8-14) q~2 and IER 2 , with the convention nln or L

for I

L~:

=00

if either

Then

106 l~nH D~nH

MALLIAYIN CALCULUS

q

q

<00 when

~ER

and, due to 8-7, the set


n.I H

q

From 8-8 we deduce

{~EH2:

is a norm.

(8-15) 8-16

DEFINITION: Let qE[2,00). Then Hq denotes the clo-

sure of R under n .II H (by 8-15 this definition coinciq des with 8-12 when q=2; note that H ~H , for q'>q). q

Set also H =n 00 2 2 q

<'"

q

H. q

PROPOSITION: Let qE[ 2,"')'

8-17

a) (H ,I ,U H ) is a Banach space. q q

b) Let {~}CH converae to ~ in Lq. Then n q Y

ll~n-~UH

+

0

~EH

(which in particular implies that

q

)

if and dnZY if {L~ n } is Cauchy in and if the sequence {r(~n'~n)q/2} is VI(= Uniformly Integrable). In this case, we aZso have r(~ n ,~ n ) + r(~,~) in L q / 2 • Lq

Proof. For q=2 the claims follow from 8-13 and 8-15. For q>2 we begin with (b). Let {~}CH go to 0 in L q . n

(i) Assume first that

q

I~

n

-~HH

q

q

+

O. Then

L~

n

+

L~

in L • From 8-7 we deduce

r(~

~ )q/2 < 2q-1[r(~ _~ ~ _¢)q/2

n' n

-

hence the sequence r(~

n

'

n

+

r(~,~)q/2J,

{r(~ ,~ )q/2} is UI. Moreover, 1n

r(¢,¢) in L n n /2 also holds in L q

n by 8-13, so this convergence

,¢ )

+

(ii)

Conversely, assume that {L~n} is Cauchy in L q

and that

{r(~ ,~ )q/2} is UI. From 8-12 we deduce that

¢EH 2 , and L~n

n

n

+

L~ in L

2

and thus L~n

+

L~ in Lq as

well. Moreover r(~ ,¢ ) + r(~,~) in Ll by 8-13, thus n n r(¢n'~n) + r(¢,~) in L q / 2 by the UI property. Since

MALLIA YIN OPERATORS

and

r{~n-~'~n-~) ~

0 in L I by 8-13, we obtain that z 0 in L q / , and n~n-~IH ~ 0 follows.

r(~n-~'~n-~) ~

q

It remains to prove (a).

I.U H . Then there exists and

107

{L~n}

{r(~ -~ n

Hence

~ -~ )q/2} m

{r{~ n ,~ n )q/2}

and U~n-~DH

~

n

Finally the family is UI,

n,m~l

be Cauchy for

q ~ELq such that ~n ~ ~ in Lq,

is Cauchy in L q .

m' n

{~}eH

Let

is UI. Thus

while

(b)

~EH q

implies that

o.

q

THEOREM: The operator (L,R o,)

8-18

1-8

a MaZliavin ope-

rator (hence, the conclusions of Proposition 8-10 hold

R).

with Roo instead of Proof.

( L , He,)

8-1-(iii)

sat i sf i e s 8 -1 - ( i i )

( b e c au s e

R eH 00 )

and

(by 8-13-c), while the restriction of r to

R~XHoo

is nonnegative (by 8-13-b). We also have H en L P , and L~En LP if ~ER . It remains to prove ~ p , .• ,~ )E(H) , FEC (R ), then 'I'=F(4)) bep

~

longs to Rco and L'I' is given by 8-3 ticular case of 8-3, we will

(since 8-2 is a par-

thus have 8-1-{iv) as

well). 1

k

Let qE[Z,co). There is a sequence {~ =(4) , •. ,4> )}eR i i n n n such that D~ -4> iH ~ 0 for all i
Since 4>n ER L'I'

n

q

n

k

n

,8-3 and 8-4 give

~

--d-- F (4) 1=1 dX i n

)L~i n

+

I

d2 Z i,j=ldxidx j

1

F(~

n

)r{4>i,4>j) n n (8-19)

108

MALLIA YIN CALCULUS

Let I;.e~o be such that

IF(x) I. IDxF(x) I,

<

e ).

Since ~ 4 ~ in Lq, the r.v. 2 n aF/ax. (;p ), a Flax .ax. (;p ) obviously converge to F(1 in L q and r(;p .
n

n

Then,

L~ n g·oes in L q/ (2+e) t

r(~n'~n)

that

n

passing to the limit in 8-19 as nt oo ,

r(~.~)

th e r. v. L~ d e f'l.ne d b y 8 -3, an d

0

also goes in L q/ (2+e)

to

=

while we have already seen that ~nER,

we see that

~n

~

4

is L q /

e.

so it follows that ~ belongs to Hq/ (2+6)'

ded

q/(2+e)~2,

8-3,

Since

q

and it also follows that

L~

But provi-

is given by

is arbitrarily large we deduce

~EH=.

§8-c. MALLIAVIN OPERATORS ON A DIRECT PRODUCT This subsection is mainly technical.

It will be used

essentially to prove the "adaptedness" of Malliavin operators on Poisson-Wiener space. We start with two probability spaces and

(n",~",p").

tor (L',R')

r'

.

H', H' ""

q

If
On each one there is a Malliavin opera-

(resp.

(resp.

(L",R")), with which are associated

r", H", H") as in §8-b. Set

(resp.

(n,&"p) =

(n',~',p')

""

q

(rl',~'.p')@(Q",~",P").

;pll)

(8-20)

is a function on n'

(resp. nil) we de-

note by the same symbol its natural extension to ;p is a function on and

n.

If

n, we denote by ;p',,(w')=iP(w' ,w") w

w" (w")=
~"

and E" for the expectations with respect to p' and P".

MALLIA YIN OPERATORS

The dipeat ppoduat Malliavin operator

8-21

DEFINITION:

(L, R)

is defined by:

R

(i)

109

is the set of functions of the form ~E1(',

where

1

'!ER" , 1

(ii) If ¢ is as above, we obviously have ¢'"ER' w and ¢"w' ER" , and we set L¢(w',w")

=

L'',,(w')

w

+ L"",(w").

(8-22)

w

PROPOSITION: The direct produot Malliavin ope~a­ tor (L,R) is a l1alliavin operator on (rl,£;,P), and the assooiated bilineap symmetrio operator r satisfies ;01' 8-23

¢,'¥ER: r(¢,'¥) (w' ,w")

r'('w'" 'II'W

II

)( W ')

+ r"(¢" w" Proof.

(8-2t.)

'¥"

w'

)(w").

L P and that P. meets 8-1-(i,ii) are p <00 8-22 obviously defines a linear operator L:

That Rcn

trivial. R ~ n

L P (compute explicit ely L¢ in terms of P <00 by 8-3 to check that L¢En L P ).

i',

'i'

p <00

That L meets 8-3 and that r, 8-24,

are also consequences of easy computations. 8-24

implies r(¢,
defined by 8-2, meets

¢,'¥ER, E(U'!')

hence (L,R)

property 8-1-(iii)

meets 8-1-(iv). for L'

Finally

and L" yields

= fplI(dwll)E'(¢~" L''Y~II) + !P'(dw')E"(¢"

w' L",!,") w'

fplI(dw")E' ('Y~II L'¢~II)

+ fp'(dw')E"('Y~, L"¢~,)

E('I'L¢).

110

MALLIA YIN CALCULUS

Let <1>': rz' "* 1R. Then <1>' EHi if and onZy if 'EH 2 • in which case L'' is a version of Lt'. PROPOSI TION: a)

8-25

b) If

<1>'

EH

i•

e " EH :; • the n r ( t ' , e")

As we shall see, (b),

0,

the necessary part of

are almost evident;

suff~cient

=

(a),

and

the non-trivial part is the

condition in (a), We begin with an auxiliary

lemma.

8-26

LEMMA:

Let ER and ~'(w')=E"(t",). Then ~'ER' and w

A

E"[ (L
L' <1>' (w' )

(8-27)

i ' .. , , e ' ) ER ,n

We have =F(i' ,i"), wher e ~'= ( 2 n d and i'''=(e ,e )ER"d and FEC (R xR ). p Proof.

1, ... d

G(x)

n

The function

= E"[F(x,i")]

(8-28)

2 n A,., clearly belongs to C (R ), and '=G(t'), p

A

hence e'ER',

We deduce from 8-22 and 8-5 that

f p"(dw")L'e',,(w') w

E"[ (U)" ]

w'

= Then,

+

E"(L"e" )

w'

fp"(dw")L'IjI',,(w'). w

since we can differentiate 8-28 twice under the

expectation,

a simple computation shows that 8-27 holds

(use 8-3 for L'), Proof of 8-25.

a)

First assume that e' EH

2,

There is a

sequence {t'}eR' such that ne'-e'u H ', ~ 0, Clearly 'ER n n n e' and L'' ~ L'' in L2(py as well as in n Since L'e'=Le' by 8-22 (recall that L"1=0), we n n obtain e'EH 2 and Le'=L'', Conversely,

assume that 'EH 2 .

There Is a

sequence

MALLIA VIN OPERATORS {4Jn}C~ such that

Then

4J~ER'

i4Jn-4J1~2 +0.

/\

Set 4J~(W')=E"[ (4Jn)~,l.

by the previous lemma,

E'(I~~_4J'12)

111

and

fp'(dw')I!pl(dwl)[~n(w"W")-oIl'(w')JI2

=

< !P'(dW')!PIl(dw ll )14J n (w',W")-oIl'(w')1 2

Since~'n

satisfies 8-27,

I

A

A

E'( L'oIlr't-L'oIl~1

2

we also have

)

!p'(dw')I!p lI (dW")[LoIl n (W',W")-LoIl m (w',w")112 < !P'(dw')!P"(dw")ILoIln(w',WI)-LoIlm(W',w")12

as n,mt=.

/\

b) Let {4>'}cR' n

n"-"II~"

n

4>' ... <1>' n

2

2

Thus {L'oIl~} is Cauchy in L (P'), and {4J"}c1(" with n

II¢'-'II~,

2

n

and <1>" ... <1>" in H2 , n

i

hence 8-13 yields 8-24

z.

... 0 and

..... O. From (a), we deduce that

41") + T(4)' 4>") in L (P). However, n' n ' that T(~,~)=O, so T(' ,<1>")=0. T(oj)'

thus ¢'EH

implies

Section 9: MALLIAVIN OPERATOR ON WIENER-POISSON SPACE

§9-a. MALLIAVIN OPERATOR ON POISSON SPACE In this subsection,

(n,~,p)

denotes the canonical space

introduced in §6-a, except that we have only the Pois-

sor, measure

~,

and no Wiener process (recall that the

intensity measure of u is v(dt,dz)=dtxC(dz), G being Lebesgue measure on the open subset E of m S ). We denote by C 2 E([O,T] xE) the set of all functions 0,

f:[ O,T]xE

~

m

that are Borel, null outside a compact subset, of class C2 on E (i.e., in the second variable) with f,D f,D22f uniformly bounded on [D,T]XE. If 2 z z fEe E([O,T] xE), we write u(f) for the random variable 0,

J).l(' jdt,dz)f(t,z).

We consider an auxiliary function p:E + [0,00) which is of class

c~

(other conditions, similar to 6-9, will

be put on p later). Set

R

the set of all functions of the form ,u(f k )), with FEC 2 (Rk) p ,

~=F(u(fl)'" 2

} (9-1)

fiECo,E([ D,T] xE). If

~

is as above, we set L~

=

1

"2

+ 2

(9-2)

112

WIENER·POISSON SPACE ~z

where

113

stands for the Laplacian on E.

PROPOSITION: 9-1 and 9-2 define a Malliavin opera-

9-3

tor (L,R). Moreover if =F(u(fl), .. ,u(f k » '¥;H(u(hl), .. ,)l(h q »

~

k

L

r(~,'¥)

and

belong to R. then

L

a

~F()l(fl),··,u(fk»

i=l j;l

a -a-H(u(h1), .. Xj

~

(9-4)

,).J(h»

).J(pD f.(D h.) Z ~ z J

q

T

).

<: L P whenever P "" f is a bounded function on [O,T]xE with compact support.

Proof. a) It is well known that u(f)En Hence Rcn

p<""

8-1-(i,ii)

L P and LEn

p<-

L P for ER. That R meets

is evident.

b) Next, we need to show that 9-2 defines a linear map on

R. In fact, the linearity will be evident, pro-

vided we prove the following: and '¥;H().J(h l ), .. ,u(h q » then,

if <1>:'11, we have

Let

IJEn

port of U. where

V

be fixed.

let ¢;F(u(fl), .. ,ll(f k )

and define L and L'¥ by 9-2;

L~:L'¥.

Let (t,z) be a point in the sup-

e

Set 1J'=u-E(t,z) And u ;U'+E( t,z+ e) for eEV, o in IRS such that is a neighbourhood of

o

z+VEE. Then U =u, and the function ~t,z(S):=

e

(u ) = F(u'(f1)+fl(t,z+e), •.• , u'(fk)+f k (t,z+8))

is of class C 2 on V. Hence 1

a C

~

(a):=p(z+e)Da~t

(e) is ,z function from V into lRll, and a simple computation

shows that

t,z

114

MALLIAYIN CALCULUS

Ik

ilF

-d-(\.I(fI)'··){p(zH, f.(t,z)+D f.(t,z)D p(z) xi z ~ z ~ z

i= I

2

k

d F L a a (\.I(fl), .. )P{Z)D i,j=l xi Xj

+

~t

Similarly, we associate

~=~ we deduce that

,z

and

/'

Wt,z

L~(].I)

}

T f 1 (z,t)D fj(z,t). Z z with~.

Since

diVe~t.z(O) = diVe~t.z(O). Summing

these equalities on all pOints (t.z) ].J yields

T

= V!'(\.l) ,

thus

in the support of

L~:::;L'!'.

c) Two simple computations show

that

L

meets

8-3. and that the associated operator r

(as in 8-2) given by 9-4. Since p>O and since the matrix (c ij =

is

T

D fi(D f.) ). j k is always symmetric nonnegative. we z z J ~, < deduce from 9-4-that r(~,~)~O: hence 8-1-(iv) holds. d)

F(\.l(fl), .. ,].J(f k » K

~

It remains to prove 8-1-(iii). Let and ~=H(\.I(hl) ... ].J(hq»

be in R. Let

be a compact set of E such that [ 0, T] xK contains the

supports of all fi's and hi's. Let

(Si,Zi)i
the points of [O,T] xK that belong to the support of ].I, with SI<,,<SN (if there is no such point. One knows

that

N

is a Poisson r.v.

then N=O).

with parameter

TG(K), and that conditionally on the a-field ted by

Nand SI •••• SN'

the r.v.

g

genera-

(Zi)i
dent with uniform distribution over K. We will prove that 1

E(~L~)=-2E(r(~,~».

between

~

E(~L'!'+tr(~.~) I~)=o:

this yields

and we deduce 8-1-(iii)

by symmetry

and'!'.

From now on, we fix

N

and the S. 's. For simpliciJ j ty of notation, we set fi(z)=fi(Sj'Z) and h.(z)= j

~

1

hi(Sj'z). From 9-2 and 9-4 we obtain that 4>L,!,+ 2f(4),'1')= g(ZI""ZN)' where

WIENER-POISSON SPACE 1

N

N

2F(Lf~(zn)""'Lf~(zn»{ 1

1

~

? i,J=l

115

a2R

N n

axiax j (lL hl (zn)"")

L p ( Z n ) Dz h.n (z n ) Dz.hin (z n ) T

N

n=l +

1 + -2

r

aH

1.

N n

N

n

-a-(Lhl(z ), .. ) L [p(z H h.(z ) n 0=1 0 z 1. n 1=1 xi I + n p(z)D hni(z )T]} z 0 z 0

N N ~ aF \ n aR \ 0 -~-(l.fl(zn)'··)-a-(l.hl(zn)'o,,) xi I j=l i= lOX j I k

L

L

n

N

L p(z n )D z f~(z )D h (zn) J 0 z 1

T

n=l

•

If GK denotes the uniform distribution over K, fices to prove that

it suf-

(9-5) Let

Then a simple computation shows that

t 0=1L N

g(zl,··,zN) =

8

L _a_Q~'~(z

~=l ax~ Zo

Since Q~,i=O on the boundary of K, z deduce tRat !CK(dz) _o_Qn,i(z) = ax. A .. zn

). 0

it 1s then easy to

a

for all n,i,zn: therefore 9-5 follows from 9-6.

MALLIA YIN CALCULUS

116

§9-b. MALLIAVIN OPERATOR ON WIENER SPACE Here we just recall, without proofs, how the proper Malliavin operator is defined on Wiener space o As a matter of fact, we give only one among many possible definitions proposed by various authors (see e.g. Stroock (24),

Zakai [29]).

It is very close to the de-

finition proposed by Shigekawa [231

(see also [12]),

and in fact even simpler.

(n,£,p) is the canonical m-dimensional Wiener

Now,

space (the one introduced in §6-a,

except that we only

have the Wiener process W, and no Poisson measure). Set

R = the set of all functions of the form kl kn 2 n 4>=F(W t •..• ,W t ), where FEC (R ), 1 n p o-

is as above. and i f 0ij=O (resp. =1)

}

if

i,;j

(9-7)

(resp.

i=j), we set L4>

= 1

(9-8)

n

L

+ -

2 i,j=1

PROPOSITION: 9-7 and 9-8 define a MaZZiavin operator (L,R). Moreover~ if 4>=F(W kl , ••. ,W kn ) and J/, J/, tl tn

9-9

'¥ = H (W 1, ••• , W q)

sl

Sq n

be Z0 ng toR. the n Q

L L i=l j=l

a

F kl -a-(H , ..• )

xi

tl

aH

~l

j

sl

-~--(w oX

"")Ok

(9-10) n

i'hj

The proof is entirely elementary, except perhaps for 8-1-(ii1)

(which follows from a classical integra-

tion-by-parts argument for finite-dimensional Gaussian

WIENER-POISSON SPACE

117

measures), and in any case is much simpler than for proving 9-3.

§9-c. MALLIAVIN OPERATOR ON WIENEF-POISSON SPACE (n,~,p)

From now on,

is the canonical space of §6-a,

with the Wiener process Wand the Poisson measure

~.

Then clearly,

=

(n,£,p) where and

(nP,~p,pP)~(nW,~W,pW),

(nP,~p,pp)

is the canonical Poisson space of §9-a

(CW,[W,pW)

Call (L

p- P

,R )

is the canonical Wiener space of §9-b. t.,r W and (L ,R ) the above-constructed ~allia-

vin operators on these spaces.

Then we set

= direct product of (LP,RP) and (L W,R W): see 8-21.

(L,R)

(9':'11)

With each these Malliavin operators we associate

r,r p ,r W and Hq , HP , HW as in Section 8. Then i f q q

are respectively ~p- and

~

~,

'l'EH2

W

-measurable, we deduce from

8-25 that r(~,'!')

= 0

]

(9-12)

We end this subsection with a property of adaptedness that is well known for

W W

(L ,R ). For tEl 0, TJ, £:t is

defined in §6-a and we set o(Ws-Wt:s~t;

of

9-14

and

~(A):

A Borel subset

(t,T]xE).

PROPOSITION: Let ~>'!'EH2 be ~t-measupabre. r(~,'!')

(9-13)

ape

~t-mea8upabZe.

Then

L~

MALLIA YIN CALCULUS

118

b) Let ~,~EH2 with $ being ~t-measurabZe, and being ~t-measurabZe. Then re~,~)=o. Proof. Since

~t

~

is contained in the P-completion of

:s
FO=Ci(W

=t

(nI'~I,PI)

the canonical Wiener-Poisson space when the

tim-e interval is an arbitrary interval

I

instead of

[O,T]. We have the obvious identification: (n[ O,t] ,~[ O,tJ ,p[ O,t])

(n,~,p)

tal (n e t , TJ '

and J;[ 0 tJ ,

(resp. 0

trace of It (resp.

f: (t

t'

I)

f: e t

, TJ ' P (t , TJ )

TJ) can be interpreted as the over n[O tJ

t '

(resp.

net TJ)' Mo,

reover let R t (resp. R ) be the set of all $ER that are measurable with respect to ~~ (resp. ~t), and let L t t

(resp. L ) be the restriction of L to R t

t

(resp. R ).

It easily follows from 9-11 and 9-1 and 9-3, and 9-7 and 9-8,

that

t

t

(Lt,R t ) and (L ,R ) are Malliavin

operators on en[ O,t] ,~[ O,t] ,p[ O,tJ) and on (n(t,TJ'~(t,TJ,P(t,TJ)' a~d ~hat (L,R) is the direct product of (Lt,R t ) and (L ,R ). Now, a function $ that is ~~- (resp. It-) measura-

ble is just the extension to n of a measurable function on (rl[ O,t) ,~[ O,t]) .(res p . (net,T] ,K(t,T]»' Hence, the claims readily follow from Proposition 8-25.

§9-d. MALLIAVIN OPERATORS AND STOCHASTIC INTEGRALS Now we seek to recognize how in §9-c, operate on a

Land

r,

as introduced

(very particular) kind of stocha-

stic integrals with respect to

U and

~

(we do not look

here for a "general theory", which undoubtedly exists: see [16]

for the Wiener case). More precisely, we con-

WIEl'.T£R-POISSON SPACE sider two

119

integrals of the form T

ft

f

f(s,~)dWS ' 0 =

t

T

JF(s,z,~)P(ds,dz) E

(9-15)

where: 9-16

(i)

¢=(~

all

(ii)

1

, ••

,~

)

is F -measurable and =t

¢iEH

for

l~i~d;

f:[O,T]xlR d -+]Rm is Borel, with

for

on lR d,

r=O,1,2,

for

r;,e~o;

F:[ O,T] XExlR d EXl1l.d,

c2

of class

ID\f(s,x)I~I;;(1+lxle)

x constants

(iii)

d

of class C 2 on

1R is Borel,

-+

and there are two constants

a function n En 2

~P~'"

LP(E,G)

1;;,8~O

and

such that

for

r=O,1,2

for r=1,2.

Note that

in virtue of Lemma 5-1,

the r.v.

ve are well defined and belong to n It is now the time to to the Cl function p: b

p <'"

~

and

5 abo-

LP •

impose further restrictions

Conditions on p: If the complement E C of E in is not empty, and denoting by of zEE to EC,

d(z,E c )

the distance

the functions

10 z p(z) 1

These conditions obviously imply 6-9-(i1) that

P, IDzpJEn l <
-p-

and also

MALLIA VIN CALCULUS

120

THEOREM: Assume 9-16 and 9-17. Then the random

9-18

variables ~ and 0 defined by 9-1£ belong to have 9-19 to 9-24 below:

R~J

and we

L'i' =

IT I{ dL

La

t

E i=1

+

2

of i -a-(s,z,~)L~

a2 F

d

L

j

i

j

I T J{p(z)n z F(s,z,~)+D z p(z)D z F(s,z,~) T }d~. E

t

r('i','I')

i

ax ox (s,z,~)r(¢ ,~ )}P(ds,dz)

i,j=1

1 +-2

(9 -2 0)

Xi

I

T

(ff

T

) (s,~)ds

(9-21)

t

r(o,a)

I t

d

+

I

i,j=l

.r('!',o)

II-' J

T

T Ip(z)(D F.D F )(s,z,~)~(ds.dz) E i

j

(9-22)

z

z

T(lF

(s , z • ~) d ~ }{ I I-a- (s , z , ¢) d il } .

r (~ ,~ ){ fJa-

tE Xi

_

TaF

tE Xj

WIENER-POISSON SPACE d

r('¥,a)

I

r( IP\ a)

d

.

i=l

I

r(c,a)

T

f

~(s "') dW

t

dX i

f f

r(IP 1 ,a)

s

(9-24)

aF

T

i=l

'

121

t E

~(s,z,¢)du. Xi

Of course, 9-21 and 9-22 allow to obtain f('!','!') and r(6,6) by polarization, if ~and

f

T

are defined by 9-15

F (meeting 9-16). Namely:

with the functions f and r( '!', 't')

7

T f(s,IP)fCs ~) ds

(9 -25)

t

d

.,

L

T

f(IP1, IP J){f i,j=l t

+

df

- - ( s
T

df

- ( s IP)dV; ax j s '

r (0,6)

+

d

(9-26) ..

I

T

r(1,J){ff i,j=l tE

of

.

T

aF

tE

Xj

-a-(s,z,¢)djjd!f -a-(s,z,
REMARK: Due to 9-16 and 5-1, all the integrals in

9-27

the above formulae are well defined. The proof of Theorem 9-18 will be broken into several steps. Firstly: 1.EMMA: Let f:[0,T1

9-28

Then '!'--=--f-Tf(s)dW t

and

f('t','l') =

f

T s

be Bore?" and bounded. betongs to Hoo' a~d L'!'=-! fTf(s)dW .... ]Rm

2 t

f(s)f(s)T ds •

s

t

Proof. This result being well known in a slightly different context [161, we just sketch the proof. Firstly if

f

then 'I'ER and the formulae

is a step function,

giving L'!' and f('I','I') are nothing else than 9-8 and

9-10 (recall 9-12). If

f

the 1 im i t i n L 2 ([ 0, T] , d t)

is Borel and bounded, it is 0

f a seq u en c e ( f n)

0

f

s t ep

MALLIA YIN CALCULUS 'T'

func t ions.

Then

'I'

f~

n

t

q <00, and l'l' .... L'I'=-'I'/2 T Tn in lq/2: f(s)f(s) ds

J

f

n

(s)dW

s

.... 'I'

in Lq

for all

in Lq, and r('I'n,'I'n) + r('I','I'):= hence 'l'EHq/2 by 8-17.

t

Next,

E

BxK

if Band K are compact subsets of E and m d

the set of all functions F(s,z,x) = F 1 (s,z)F 2 (X), where FlEC 2 E([O,T]~E) has its 0, 2 d support in (t,T]XF., and F2EC (R ) has its support in K.

E'BxK

the linear space spanned by E BXK '

E"BxK

the set of all functions F on [O,T] xExm of 2 d class C on Exm , with support in (t,T] xBxK, 2 2 with F,DzF,D 2F,DxF,D 2F uniformly bounded. z x the set of all functions f(s,x)=f l (s)f 2 (x) where fl is Borel bounded with support in (t,T]

F

K

d

and f2EC2(Rd) with support in K.

F'K

the linear space spanned by

F"

the set of all

K

[0, T] xm

d

9-29 OEH

'"

mm-valued functions 2

of class C on m

(t,T]xK, with

FK .

2

f,Dxf,D 2f x

d

f

on

, with support in uniformly bounded.

LEMMA: I: f~fEFK and F,FEE~xK we have '1', '1', c,

and 9-19 to 9-26 hold.

ProoL All equations 9-19 to 9-26 are linear,

except

9-21 and 9-22, which indeed follow from 9-25 and 9-26 upon setting f=f and F-r. So we can suppose that f= f 1 0f2 and 7=7 l &f 2 are 1n F 1 0F2 are in EBxK

T

and that F=F 1 0F 2 and F=

T ~'=[ 7 l (s)dW , which are in Hoo t s t _ s_ Set o'=u(F1)-V(F 1 ) and c'=U(F1)-V(F 1 ), which

Set '1"=[ fl(s)dW by 9-28.

FK

WIENER-POISSON SPACE are in H", (and even in R)_

123

By 8-18,

f 2 (4)), £Z(1)), and F Z (1)) are in H",. Then ~=~'fZ(1)), ~=~'£Z(1)). o=o'F Z (1)) and 6=6'F Z (1)) are all in H",. We prove 9-Z0, for example. hence 8-3,

F Z (
r(o',F Z (1»)=O by 9-14,

9-Z and 9-12 yield

Lo = o'L(F Z04»

+ F Z (4))Lo'

d

il(F 1 ){ 1

+"2

L

_o_F (¢)11>i i=l dX i Z

aZ

d

L a a

i,j=l xi Xj

F Z ( 1» f (1)

i

j 1 , 1> )} +-2F2(1))~(p6 z FI+D _ z p. T

DzF l ), which is nothing else than 9-20:

9-19 is proved similar-

ly (use 9-28). We also have r(0,6)

= o'6'r(F Z (1»,F 2 (4») d

L

i,j=l

+ F 2 ($)F 2 (¢)f(C',5')

il(Fl)il(Fl)~Fz(
Xj

_ -T + FZ(1))FZ(¢)~(pDzFl·DzFi)

(use 9-4,

9-1Z,

9-14, 8-4), which is nothing else than

9-26;

9-25 is proved similarly_

9-1Z,

hence

r(~,o)

f(~',c')=O

obtains from

= '!"O'r(F Z (¢),f Z (1») d

HZ H2 .' L ' (¢)-(1»f(1>1.,1>]) X

i,j=ldx i

a

j

which is 9-Z3. Finally, 9-Z4 is proved similarly. 9-30

LEMMA:

Let

FEE

BxK :

Zet

finite measure on [O,T] XEXlR d ;

qElN*

and y be a positive

Zet B' (resp. K') be a

compact neighbourhood of B (resp. of K). There is a sequence {Fn}cEi'xK' such that

Z Fn' DzF n , DzzF n • DxFn'

MALLIAVIN CALCULUS

124

D22F ccnverae respectiveZu to F, x n ~ d ~ LqC[O T]xExJR ,y)

DzF,

in

We can obviously suppose that E=JR B

Proof.

a) We first

show that it suffices

sult when FEE~xK is COO in (z,x).

compact neighbourhood of BXK, 0) S+d terior of B'xK'. Let (cp )CC (R ) n

0

and set

(the convolution product). belongs to its first for F,

EB"XK"

prove the re-

For this,

be a

sequence of functions,

to

contained in the inbe a

regularizing

Gnes,.)

Then G

n

let B"xK"

= F(s, .)*4>n

is C'" is

(z,x),

it

and Gn and and second derivatives converge to the same for all n large enough,

in Lq(y). So we may suppose that F

b)

is C'" in (z,x).

Set

a2(S+d) G(s,.)

There is a

=

2 2? 2 F(s,.). dz l ·· .dZSdxi·· .dx d

sequence of Borel functions

Gn(s,z,x)=

Ll <· < k ~

g n, i(s,z)g n, i(x), continuous in (z,x), such G -> Gin Lq(y), with g ,(s,z)=O if s
th;t- n

QO

if h

which equals K'c) and

1

on

B

Crespo K)

and

0

on B'c

(resp.

if

k

n

I

h(z)h(x)

i=l xU du J dv g ,(s,v)} { u I.' ~z i } {v ,
xU

dy

{Yi~xi}

(wherde z=(zi)' in lR

),

v=(v i )

J

dy' gn,i(y')},

{y~~Yi} are in JRS

,

Y=(Yt),

Y'=(Y~)

the sequence (Fn) meets our claims.

are

WIENER-POISSON SPACE

125

There is of course a similar result about

" K'

with

a simpler proof.

9-31 p.~,

LEMMA: Let fEF~ and FEE;xK.Then 6 and' beZong to and 9-19 to 9-24 hoZd.

Proof.

Let B'xK' be a compact neighbourhood of BxK.

Y.p be the law of .p on JRd,

Let

and v'=l(t,TlxB"v, and

Tl (s)ds, and set y = v'ey ... and y = Y'~Y.p t, d'" d (finite measures on [O,TlxExJR and [o,TlxJR ). Let

y'(ds)=l(

q>4 be fixed.

}eE', , be a sequence approaching F_=F in n B xK Lq([O,TlxEXlRd,y) in the sense of 9-30; let {f }eF', Let {F

d n

be a sequence approaching f",=f in Lq([O,TlxlR

,y),

K

again in the sense of 9-30. d

For a function G on [O,TlxExJR (resp. g on d n T n [O,T]xlR ) we write !l(G )=/ r Gn (s,z,4»dll and simin t t T larly for v(G ) and n(G) (resp. w(g )=( g (s,4»dW Tn n n i: n s and O(g )=/ g (s,q,)ds), provided those random variables n t n are well-defined. Then we deduce from 9-30 and from the very definition of y and

y

that

F , (IF lax., n n ~ or p~ z F n +D z p.D z FTn (resp. if g n functions fn' (lfn/ax i , or of the functions

9-33

Now,

Holder's

inequality yields for

p~l:

126

MALLIAVIN CALCULUS

E[ !J«t,T] XB,)p-1/2 u(ICn-GcoI2P)1/2]

<

E[uCIG n -G oo I 2p )]1/2

< E[)J«t,T]xB,)2p-1]1/2

c

EC\lcIG -G 12p))1/2 n co

p

(the last equality, where c

is some constant, comes

p

fromthe~®~-measurability

Thus u(G n )

~

of GnCs,z,
wCG oo )

Crespo in 9-33), and by difference U(G n ) ~ in L q / 2 (resp. L q / 4 ) as well. We also deduce ea-

in 9-32

P(G oo )

sily (from Lemma 5-1 for in t q (resp.

instance)

that w(gn)

i f gn is like in 9-32

L q/2)

~

(resp.

wCg",) in

9-33). Hence 'Y

....

:= w( f n)

n

:= iiCFn)

<5

n w(afn) ax.

~

l.

of n) itC-aX i u(p~

~

:= w(f) ,

'Y

:= it CF) , a 2f Of n) w(ax:), w( oxiax j l. a 2F _ C dF ) j.J~, it ( ax. an) x. 1. i J ~

<5

T

z

Fn+D p.D F ) .... z zn

u(p~

.... ~

we

a 2f ) dxiax j

,

a2 F P(ax.ox.)' l. J

\

(9-34)

T

z

F+D p.D F ), z z

all these convergences taking place in L q / 2 and

(9-35)

We deduce from 9-29 that

~n,QnEH",

and we can write 9-19

to 9-24 for 'Pn,on' which gives for nElN: af

d

L'I'

n

L

W (

----.E. ) L i

i=l aX i

1 + -

d

L

2 i,j=l

WIENER-POISSON SPACE

Lo

127

n

il(pD F

z n

.D F

T

z n

)

(9-36)

+

f

d

r.

i, j =1

I

I

)

Moreover,

if we define

L~,etc

I

... by 9-19 to 9-24,

random variables are also given by 9-36, with

those

f,F

,F . Since L~i, r(~i,~j), r(~i,a)En LP , n n p <00 we readily deduce from 9-34 and 9-35 that L~n ~ L~, instead of

Lo

n

f

~ Lo in L q / 2 -£

r(on,on)

~

and r(~n'~n) ~ r(~,~),

r(o,o), r(, ,0 )

and r(on,a) ~ r(o,a)

n

in

n

Lq

~

/4

r(~,o),

r(~

,a) n

-£,

all this for arbitra-

ry £>0. Then 8-17 yields that' and 0 belong to H /2 q

and that 9-19 to 9-24 are valid. Since

q

-£

is arbitra-

rily large, we obtain the result. Proof of Theorem 9-18. For simplicity, we set d(z)= d(z,E c )A1, where d(z,E c ) is the distance between zEE and the complement E C of E in lRB For each nElN*

we set

(and d(z)=l if E=lR 8 ).

128

MALLIA YIN CALCULUS

K

{xEJR d :

n

1x I

~n},

C

n

1

C nD . n n

{zEE: d(z»-}, -n

D n

Kn and Bn are compact. Observing that in case EC~0 the distance between D

and (D Z )c is l/Zn, d

n

obtain functions ¢n:JR

n

-+

it is easy to

and 1}!n:E -+ [0,1] which

[0,1]

are c'Z and satisfy i f xEK d>

n

(X)=r

,0

-1

1~ n (z)={ 0

n

i f x~KZn'

i f zEB

n

(9-37)

i f z~B2n'

and

ID 2 z q,n l

1Dxq,n

l

10 1~

(z)1 -< c(l + nlD

z n

..::. c,

Z ID 2!Pn(z) z

I

~ c(l+n

=

2

...... D (z) ) 2n n

In . . . . 0 2n

for some constant fn(s,X)

< C,

x

c

1

(9-38)

< c(l+d(z» 1

< c(l+---2) d(z)

(z» n

not depending on

n. Then we set

f(s,x)$n(x)

(9-39) Fn(S,z,x)

=

F(s,z,x)1}!n(z)$n(x).

By construction, f n EF K and FnEE~ 2n . define ~n and on by 9-15, w1th fn an of f and F, Lemma 9-31 applies.

an

xK 2n

In particular,

and, with the notation of the proof of 9-31

\l(G n ), If

w~

,so if we . Fn 1nstead ~n,6nEHoo

(namely,

»,

D(G n ), w(g n ), ~(g n equations 9-36 are valid. can show that 9-34 and 9-35 hold for all q
end of the proof of 9-31 will give the claimed results. It thus remain to prove 9-34 and 9-35. Let n(z) be as in 9-16;

d(z)~l

~'

(depending on

~,c,n),

and sup

and

n(z)
since

e

WIENER-POISSON SPACE

Int" f (s,x)1 xr n

<

IDr Fn(s,z,x)1 xl."

~'(l+lxle),

129

r=O,I,2

\

< ~'(l+lxle)r)(z), r=0,I,2

C9-40)

I

J Let also

Hs = 1;'(1+I41!9)1 Ct ,Tj(s), which is predicta-

ble and has: for all p <00.

(9 -41

)

2

Let g n be f n or af n lox i or d f n lax.ax., and accor~ J dingly for g. Then 9-37 and 9-39 yield gn -> g pointwis e , wh i I e 9 - 4 0 Yi e Ids

I g n (s,

4»

I -
deduces from 9-41, 5-1-b and from Lebesgue dominated w(gn) ~ w(g)

convergence theorem that

in Lq for all

q
n

n

-7

~(g) in L q

f or all q <00. 2

Let Gn be Fn or aF lax. or a F 13x.ax., and accorn ~ n ~ J dingly for G. Then 9-37 and 9-39 yield Gn -7 G pointwise and 9-40 yields IG n (s,z,
-7

G

pointwise. Moreover 9-40 yields Ie (s,z,~) I ~ 2

2 n

(H s +H s )n'(z), where n'(z)=o(z)/d (z) + ID z p(z)l/d(z) belongs to n l

< E[ sup -7

TIH +H2IPCll'l\-\lr)p] s< s s

<

00

)J(G) in Lq for all q
At this point, we have proved that 9-34 and 9-35 hold for all q
Section 10: MALLIAVIN OPERATOR AND STOCHASTIC DIFFERENTIAL EQUATIONS

§lO-a.

THE MAIN RESULT

In this whole Section 10,

the setting is the same as in

§6-a. The auxiliary function p:E

10-1 con.dition on

p:

(i)

p

+

[0,00) meets:

is of class C~, with

IDr pIELl(E,G) for all rElK zr

(the same as 6-9-(i».

(ii) Condition 9-17 is met

(recall that this is

stronger than 6-9-(ii». ~ith

this fur.ction,

the Malliavin operator (L,~)

is as

defined in §9-c. We consider Equation 6-2, with a fixed initial condition xO' We assume something slightly stronger than (A'-r): 10-2

ASSUMPTION (A'-r): The same as (A'-r)

except that n En 2

2P~()()

(see e-3),

LP(E,G).

Our aim in this subsection is to prove the: 10-3

t~T

THEOREM: Assume (A'-3) and 10-1.

Then for ea~h

the aomponents x~ of the solution X to 6-2 beZong . .

to Hoo. Moreover i f u~J

i'

..

= r(Xt'X~) and V~=LX~, th~re are

versions of the pl'oaesses

U= (Uij) . . . . . and V= (V~). d ~ ,J <" ~<

that are soZutions to the following Zinear equations:

STOCHASTIC DIFFERENTIAL EQUATIONS

131

+ [U D a(X )T+D a(X )U l~t -

X

m

+

L [u

i=l

-

X

-

-

D b·i(X )T + D b·i(X )U ]JfW i - x x --

(10-4 )

+ [U D c(X )T + D c(X )U l,*ii -x

-

x

-

-

m

L Dxb·i(X_)U_Dxb·i(X_)T~t

+

i=l + Dxc(X_)U_Dxc(X_)

T

*~,

v

(10-5)

+ D a(X )V *t + D b(X )V *W + D c(X )V x

--

x

--

x--

*0.

To prove this theorem, we begin by introducing the Peano's approximation to 6-2, namely Xn = Xo + a(X n )*t + b(X n )*W + c(X n )It-O ¢n ¢n ¢n

(l 0- 6)

where ¢n is defined in 5-28. Then 6-2 and 10-6 are of type 5-3 and 5-27 respectively, with An=A=a, Bn=B=b, cn=c=c. So assumption 5-29 is readily verified, with z~n=O,

and from Theorem 5-31 we deduce that 10-6 has a

unique solution Xn, and that I

(Xn_X)* U + 0 T LP

for all p<=.

Next, for each nElN*" process:

(l0-7)

we define the following didUig

132

MALLIA VIN CALCULUS

Dxa(X n n ) (t-$~) + D b(X n ) (W -VI n) ~t x $~ t $t

(10-8 )

t

+

f J

D c(Xnn,z)p(ds,dz). $n E x $t t

LEMMA: For every pEr 1,00) there is a constant Cp '

10-9

not depending on n. such that for aZZ t
d

M =dl+ •• +d ,when lRdis graded as lR d = s s

lR lx ••• xm q for the grading appearing in

(A'-3».

i .

Proof. Let a J(x,z) be any of the functions i ik 1 i aa lax. (x), or ab lax j (x), or -(-)ac lax. (x,z). By (i) J i' n n z J e of (A'-3) we have la Jex n,z)I~~el+lxnnl ). Then CPt $t 1 10-8 and Lemma 5-1 yield a constant 15 independent on p

t and n, with

n

t-$t~T2

-n

. Then 10-10 follows.

If moreover i,j are like in 10_11, by (iv) of (A'-3)

we have

laij(x;~,z)15.

Then if BEF n we obtain =$t

from Lemma 5-1: . E(1

B

IKn,ijI P ) t-

<

ol~P/E[(1 P

$n

B

)P]ds =

clr;ppeB)(t-~). P

t

This being true for all cl~PT. P p

15

10-12

BE~$n'

we deduce 10-11 with

t

LEMMA: Let nElN*. ~le have Xn,i EH t

CD

fo!' all i_
STOCHASTIC DIFFERENTIAL EQUATIONS

n iJ·

andU' t

n i

n J.

n i

=r(x' X')andv' t' t t

133

LXn,i are solutions t

of 00-13 )

1 n 1 n n T - -2b(X )~W+-2[P~ c(X )+D pD c(X ) ]*~ IP n z IPn Z Z IPn

+D a(X n )V n ~t+D b(X n )V n ~W + D c(X n )V n ~p. x $n $n x $n $n x $n $n n n i n n Proof. Since ~O=xo we have XO ' EH~ and UO=O, VO=O. Suppose that Xn ,1 EH for some kElli and all 1~do Let kT2 -n ~ -n -n n-n tE(kT2 ,(k+1)T2 ], hence $t=kT2 , and 10-6 reads xn,i t

Xn,i + ai(X n n ) (t_$n) + ftbio(Xnn)dW ~n $ t ~n IP s 'l't t 'l't t tin + (X n,z)P(ds,dz), $n E $t

J Jc t

134

MALLIA YIN CALCULUS

~i and oi the first and the second integrals

and call above.

By 8-18 we have a

by 9-15, with f(s,x)=b i

c

(x,Z)l(4>~,~ (s).

i

i

n

• ~

(X n)EH

and 0

i

are given

'"

t '(x)1(~n 'f

i

t ,t1

(s)

and

F(s,z,x)= n

These functions meet 9-16 with 4>t

instead of t by (A'-3). Then Theorem 9-18 yields that i i n i . '!' ,- 0 E H00 : t h us X t' E Hoof 0 r a I l ~.2d. T his is t rue in p articular for t=(k+l)T2- n : then an induction on k i

n

shows that X t ' E Hoo for all t
i

I

vn,i = Vn,i + (t_$n) ~(Xn )Vn,j t 4>~ j=l -t aX j 4>~ ~ l d a i ok 1 + 2 (t_4>n) a (X n )U n ,] j , k=l t aXj dxk 4>~ 4>~

L

abi. 1 d ,,2bi. ok _(Xn )V n ,] +a (X n )U n ,] j=l dx j
Jt { dL

+

~

I

- l. b i.(X n


2 d \'

1 + 2

L.

j,k=l

+1. Jt j{p(z)4 2 ~n E 'f

t

obtain

+ s

a2 c i a x ax j

z

J J{ I

~

E

i

~(Xn ,z)Vn,j j=l aX j ~ ~

n n,jk _ (X n' z )U } \.I (d s , d z )

k

~

ci(X n ,z)+D p(z)D ci(X n ,z)T}\l(ds,dz). n z z n t t

Recalling that v~=o, Next,

d

t

)}dW

this easily gives 10-14.

using 8-4 and 9-23,

9-24,

9-25 and 9-26, we

STOCHASTIC DIFFERENTIAL EQUATIONS

135

+

+

L

k,R.

t

+

L { Icpn

k,R.

t

abi. (X n )dW aXk cpn 5 t

t acj(X n z)dIn ax .pn' IJ cp E ~ t t

t j + In ab · (x n )dW aX k 5 .pn .pt t

I

t

i

I I .pn t

~(Xn ,z)dP}Un,k~}

E aX~.pn t

.pn t

So we have: (lO-IS) t

+

I I

.pn E t

136

MALLIA YIN CALCULUS

t

i j n + un,kj ~(Xn )}ds L{Un,ik a a ax-(Xq,n) q,n k q,n aX k q,~ k t t t t t j n ) + J L{Un, j k
J q,n

+

Jn J

t

E


t d

L

+

k,R.=l (recall that Un is a symmetric matrix, and Kn is given by 10-8). Then Ito's formula yields

Plugging this into 10-lS yields 10-13. Before proving Theorem 10-3, we still need one more auxiliary result. We denote here by

~X

the process

~xxO

introduced in 6-30, and solution to ~x

=

I + DxCX_)VX_*t +

DxbCX_)~X~~W

(10-16) + Dxc CX_) ~X_~P

STOCHASTIC DIFFERENTIAL EQUATIONS 10-17

LEMMA: Assume (A'-p)

137

for some r~3~ and caZZ U

and V the solutions to the linear equations 10-4 and 10-5. Then (X,VX,U) and (X,U) are soZutions to equati,v

ona of type 6-2 satisfying (A'-(r-1)), and (X,VX,U,V) and (X,U,V) ape solutions to equations of type 6-2 satisfying (A'-(r-2)). Proof. Let X=(X,VX,U,V), which takes its values in F= md, where d=2d+d 2 j the points of F are denoted (x,y,u,v).

Putting together 6-2, 10-16, 10-4 and 10-5,

one sees that X satisfies an equation 6-2, with initial condition (xO,I,O,O) 10-18

10-19

and coefficients

If d
-i a (x,y,u,v)

d

m =

d

i

(x, z ).

d

:

aa j ( x ) YH , -,,-

I

£=1 "x£

Clb js R.k a;-(x)y

4 l.

.t= 1

c

c

corresponds to the (j,k)th

®lR

-is b (x,y,u,v) = -i

given by:

i

-i

If i~d, a (x,y,u,v) a (x), -is is-i b (x,y,u,v) = b (x), c «x,y,u,v),z)

component of

«x,y,u,v) ,z)

~L £=1

10- 2 0

(~,b.c)

£

Clc j

-a-(x,z)y

H

xR,

If

d + d 2
a

(x,y,u,v) =

I 5=1

+ d

+

L

£=1

bjsbks(x)

MALLIA YIN CALCULUS

138 d

j s

m

L ~(x)u.tr

L

+

.t,r=l s=l ax.t +

-is b

(lb ks

--(x) ax r

~L

a cj .tr ac k -a--(x,z)u a-x(x,z)dz E .t,r=l xi r

J

(x,y,u,v)

ci«x,y,u,v),z) = p(z) +

d

L {u

ac k -a (x,z)

j i

d

+

10-21

L .t,r=l

+

xi

R.=1

ac j R.r ac k -a--(x,z)u ax-(x,z). x.t r

I f d+2d 2 <1
z

E

-is b

+

(x,y,u,v)

~

abj

5

z

z

2

L --a--(x)v i=l x.t

i!.

-

1

2

b

j

s(x),

-i

c «x,y,u,v),z) +

ac j .t 1 . -a--(x,z)v + -2{P(z)1l cJ(x,z)+D p(z) i!.=l x.t z z. T D c J (x,z) }. z d

L

Now, we grade F exactly as in the proof of 6-42, and then a simple examination shows that these coefficients are graded according to this grading, and that they fulfill

(A'-(r-2». Since 10-20 and 10-21 do not depend

STOCHASTIC DIFFERENTIAL EQUATIONS

on

y,

139

(X,U,V) satisfies also an equation 6-2 with

(A'-(r-2». Next,

(X,VX,U) satisfies again an equation 6-2 who-

se coefficients are graded, and since there is no second derivative in 10-18,

10-19, 10-20, these coeffici-

ents fulfill

It is the same for

(A'-(r-l».

(X,U), for

a similar reason. /'.

Proof of Theorem 10-3. We have just seen that X=(X,U,V) A " A satisfies an equation 6-2 whose coefficients (a,b,c) ~ "'n n n n meet (A'-I). Let X =(X ,U ,V). A close comparison bet-

ween 10-4 and 10-13, and between 10-5 and 10-14,

shows

that Xn satisfies an equation 5-27 with the following coefficients: -

2 An i Ai If id+d : A ' (y,w,t)=a (y), ~n,i5 _"'is An,i _Ai B (y,w,t)-b (y), c (y,w,t,z)-c (y,z).

- If d
'

Ai (y,w,t) = a (y) +

"n is B'

(y,w,t) js + ~(x)u~r Kn,kr(w)}, ax ~ t-

,.. n i

C '

Ai

(y,w,t,z)=c (y,z)+

~

L

n j ~

{Kt~

(w)u

R.r

ac k a;-(x,z)

~,r=l

r

+

,....

ac j ( x,z ) u R.r ax

Kn,kr( )} tw.

R.

Since (~,~,~) meets (A'-I), we easily deduce from what

140

MALLIA VIN CALCULUS

"n An I'n precedes that (A ,B ,C ) satisfies 5-29, with

r:IKn,i j I -

for some constant sup n Illz nil ~

<00

~.

t-

'

In view of Lemma 10-9, we see that

and II ~ I ~ I ~ ..; 0 and

Bp <00

if

Bp is defined

by 5-30. Hence 5-31 yields that

n i i n i n j ij In particular, Xt ' + X t , r(X t ' ,X ' ) + U and . . t t LX n ,1 + V~ in all LP . Since X~,iEH~ by 10-12, we det i i j ij i i duce from 8-17 that XtEH oo and f(Xt,Xt)=U t and LXt=V t .

§10-b. EXPLICIT COMPUTATION OF U i

j

the matrix f(Xt,X t ) plays a particularly important role. So we proceed to

As seen in Section 8 (see 8-10),

"explicitely" compute the solution to 10-4. From now on, we make explicit the dependence of X xO upon the initial condition, writing X for the solution to 6-2. Then the solutions to 10-4 and 10-5 also

xo

depend on x o ' namely U

xo

and V

. We also write VX

Xo

for the solution to 10-16 (or 6-30), and we use the notation introduced in §6-d, namely TX n

and VXn(n)

UX =VXx(n) t

t

t

+JJ ETx

x

{U X x Tn - l

VX (n)

n-l

(see 6-31),

(see 6-35).

Then for T~_l.2t
PROPOSITION: Assume (A'-3).

10-22

KX

-1

s-

t

+

Jx VXx(n)-lbbT(Xx )VXx(n)-l,T ds sssTn-l

[(I+D c) x

-1

T

D cD c (I+D c) z z x

-1 T

'1 (x

x ,z) s-

vxX(n)~=,Tp(z)lJ(ds,dZ) } vxx(n)~

141

STOCHASTIC DIFFERENTIAL EQUATIONS

Proof. In the proof we drop the superscript "x". Let us call R (=Rx) the process defined in 7-5, with only one Poisson measure

~a=~'

By 7-1, R is well defined, with

finite variation. Moreover,

VX(n)=l is well defined on

[O,Tnn and is left-continuous, so the following makes Tn_l~t
sense for

U

t

= U

Tn _ l

+

J

t

VX(n)-l dR s-

Tn _ 1

(10-23)

s

Furthermore, recalling 2-10 and 7-5, we observe that the right-hand side of the claimed formula is, for T n _ 12t
(10-24)

U(n)t Next, we set

(10-25) (again this is well defined, because of (A'-3) and the property pEL 1 (E,G». Then, comparing (10-25) and 7-5 and 10-23, and using the definition 6-31 of K (=K x ), we easily obtain for Tn _ 12t
J

+

t

Tn _ 1

Vx(n)~~ (I+6K s )-1

(10-26)

dH (I+6K )-l,T VX(n)-l,T. 5

5

5-

Moreover, we deduce from 10-4 and 6-31 and 10-25 that, still for Tn _ 12t
Ut

= Ur

n-1 m

+

~

i=l

+ (Ht-H T

) + n-1

f

t

Tn _ 1

(U

s-

dK T + dK

s

s

u

s-

)

t

JT

n_l

(10-27)

142

MALLIA YIN CALCULUS

Now we can apply Ito's formula to the triple product in 10-24, on the open interval]T getting for T

l
n- -

l,T [, thus nn (and with [Z,Z')c denoting the

n

continuous part of the quadratic co-variation process between Z and Z'): D{n)t -

DT

n-l

+ VX(n)

s-

J

+ dO

T s-

t

n-l

{VX(n) .

ij 5-

d(VX(n»T 9

g-

VX(n)T + d(VX(n» s-

S

ij

5-

VX(n)T} s-

d

L

+

i, j-l

+ L.\'T

n-l

<5 <

t{ [VX(n)

-

5

+ll(VX(n»

9

) (ij 5- +llU) 5 [VX(n)

-

VX(n)

-

VX(n)

ij

5-

g-

5-

VX(n)T - VX(n) s-

llU VX(n)T 9

-

5-

ij

S-

5-

ll(VX(n) U 5

g-

J

T +

J

5-

s-

5

VX(n)T}. 5-

Tn_l~t
dK T + dK U(n) S

5

5-

}

n-l

t

bbT(X

Tn _ l

m

+

{D(n)

5

ll(VX(n»T

Then 6-35, 6-31, 10-25 and 10-26 yield for t

+ll(VX(n»)T

g-

)dg + LT n-

1

<

5

<

t(I+llK )-lllH (I+llK )-l,T S

5

S

t

L J i-I T

n _1

+ LT

n- 1

<5
{llK (I+llK )-l 11H (I+llK )-l,TllKT 5 5 5 5 S

+llK (I+llK )-lllH (I+llK )-l,T + llK U(n) 5

S

S

5

S

5-

llKT} 5

STOCHASTIC DIFFERENTIAL EQUATIONS t

J

T m +

I J

{U(n)

s-

n-l

dK

143

T + dK U(n) } + (Ht-R T s s s-

) n-l

t

D bi,(X )U(n) D bi,(X )ds x ss- x s-

i=l T

n-l +

LTn _ l <s~t

AKs U(n)s_ AK;

(we use the property t +

This is a linear equation in to Equation 10-27 in

U:

J

bb

Tn _ l U (n) ,

T

ex s- )ds),

and it is idBntical

hence U(n)t=U t for T l
§10-c, APPLICATION TO EXISTENCE AND SMOOTHNESS OF THE DENSITY This subsection is similar to §6-f,

in the sense that

we prove here and there almost the same results, but presently with Malliavin-Stroock's approach instead of Bismut's one, Firstly, we know that

(L,H oo )

is a Malliavin opera-

tor (see 8-18) and we have described in 8-10 a related integration-by-parts setting for every random variable that belongs to Hoo' More precisely, under (A'-3), the x x x x following (crt,yt,Ht,ot) is an integration-by-parts setx

ting for Xt : x

x

Yt = -2V t , oX,j('f) t

=

i f 'fEH co ,

} (10-28)

In order to apply the results of Section 4, we still have to describe the sets C~ O(q)

,

it

introduced in

(see after 4-19), Consider the process

144

MALLIA YIN CALCULUS

(so Y~(O)=Y~(l», which by 6-42 is well defined for all q~r-l, under (A'-r) with r~3. Then C~ O(q) ,

x

is the set x

of all components of YO(q) at time t, say Yt O(q), and , the iterates CX .(q) are defined by 4-10. Recall that t ,J

yX j (q)

is a mult i-dimensional variable whose compo-

t,

n en t s con s tit ute CXt . (q) • ,J tv

10-29

LEMMA: Assume (A'-r) for some

r~5.

Then

a) {X~} is (r-l) times F-differentiable. x b) Ct,r_4_q(q)CR", for

1~q~r-4,

an d Cxt ,r_5(0)CH",.

c) x~SUPt
o

Proof. Set Z'

x

x = YO(l), as defined above. Lemma 10-17

yields that ZO,x satisfies an equation 6-2 with ~ 0 x (A'-(r-2». Since r~5, the components of Zt' are in R", by 10-3, and are F-differentiable in x by 6-29, and 1 x a x a x a x a x we consider Z ' = (Z ' ,liZ' ,r(Z ' ,Z ' » . Then by Lemma 10-17, Zl,x satisfies an equation 6-2 with (A'-(r-3».

If r~6 we can pursue the construction, as

such: if Zk,x is defined, and if it is a solution of an equation 6-2 with (A'-(r-k-2», and if k~r-5, then by 10-3 and 6-29

zk,x is F-differentiable in x and its k+l x components are in R",: so we may set Z ' k (Zk,x,llz ,X,r(zk,x,Zk,x», which satisfies an equation

~-2 with (A'-(r-k-3» Now,

from Lemma 10-17.

(a) has already been proved, and (b) and (c)

immediately follow from what precedes, once noticed that Y~ O(q) is just a subfamily of the components of q-l x • x Zt • , so an induction shows that Yt,j(q) is a subfa-

STOCHASTIC DIFFERENTIAL EQUATIONS

j+q-1 x mi1y of the components of Zt ' (for zj,x (for q=O) t

q~1),

145

or of

(for (c), the argument is the same as at

the end of the proof of 6-47).

10-30

THEOREM: Assume (i'-j) and 10-1, and set

Q:

det(U~), where UX is the solution to 10-4~ and (=~

if P(Q~=O»O)

(10-31)

a} If j~4 and Q~+O a.s., x~ admits a density Y~Pt(x,y).

b) Moreover Pt(x,.) is Of alass c r , provided: - either j~r+d+5 and q~(2r+2d+2+£)<~ for some £>0, - or j~r+5 and q~(2d(r+1)+£)<m for some £>O~ a) Morffover (x'Y)~Pt(x,y) is Of class c r , provided: - either j~r+2d+5 and X",-? q~(2r+4d+2+e:) is ZoaaZZy bounded for some £>0, - or j~r+5 and x 4q~(4d(r+l)+e:) is "Locally bounded for some e:>0. d) Moreover, (i) If j~2r+4d+8, if sup

EA q~(4r+8d+8+£)<m

for every bounded subset A and some £>0 (depending on A). and iF t>to'x

Ide t [ I +v D c (x, z)] I ~ C;

x

¥v E [

0,1]

(10-32 )

for some aonstant C;, then (t,x,y)_ Pt(x,y) is of alass Cr on (to,T ] xlR d xlR d • x

(ii) If j~2r+6, if sup> EA qt(4(r+1)(2d+l)+£)<m t tQ,x for every bounded set A and some £>0 (depending on A).

MALLIA VIN CALCULUS

146

and if c-=O. then (t,x,y)_> Pt(x,y) is of aZass d

c r on

d

(to,r]xlR xlR •

Proof.

Ca) follows from Theorem 4-7, once noticed that

under (A'-4)

the process (XX,U X )

6-2 with (A'-3) by Lemma 10-17.

satisfies an equation (b),

(c),

(d) follow

from Theorems 4-19, 4-21, 4-31, plus Lemma 10-29 and Lemma 6-51. 10-33 REMARK: Compare this with Theorem 6-48: of courx se Qt is not the same variable in both theorems, but we x x x shall see that the estimates on qt(i) when Q =det(U ), in the next section, are the same as they were when QX=det(DX x ) in Section 7. However, one needs one more (resp. two more) degrees of differentiability on the coefficients (a,b,c) in 10-30-a (resp. 10-30-b,c,d) than in 6-48-a Crespo 6-48-b,c,d). Furthermore, we need 10-1 (stronger than 6-9), and (A'-r)

instead of (A'-r). Hence Theorem 6-48

is (slightly) better than Theorem 10-30.

Section 11: PROOF OF THE

MAI~

THEOREMS VIA

MALLIAVIN'S APPROACH

§ll-a.

INTRODUCTORY REMARKS

We want here to deduce Theorems 2-14, 2-27, 2-28, 2-29 from Theorem 10-30. And,

exactly as in Section 7, we

need to extend the setting of Sections 9 and 10 to encompass the situation of Section 2. So, we supporting

cons~der

the canonical setting of §7-a-l,

W'(~a)a_~'~'

As in Section 10, the needed

regularity conditions on the coefficients are slightly more than (A'-r), namely:

11-1

ASSUMPTION (i'-r): The same as

except that D en 2 For each Ea

+

[O,~)

Now,

(A'-r) in 7-1,

LP(E ,G ) for all a=l, ... ,A.

a

2.P~ClQ

a

a

a~A

we also consider a function Pa:

satisfying 10-1 (and thus 9-17 as well).

for translating Sections 8 and 9 the most con-

venient way consists in aggregating all measures J.!

~a

and

into a "big" measure Ii=L~a+~' which is a Poisson mea-

E=LE

sure on [O,T)XE, where

a a

+E ("disjoint" union).

Then one considers n as being the canonical space accomodating Wand Ii. And the auxiliary function

P

which

serves to constructing the Malliavin operator is P

=

0

(11-2)

on E.

Obviously, all of Sections 9 and 10 carries over without modification, with Wand Ii. back to the original measures J.!a and 147

If we then come ~,

the fundamenta 1

148

MALLIA YIN CALCULUS

formula 10-4 becomes UX = bbT(Xx)lt-t +LP -

CJ.

(D c ) (D CJ.

Z

Z

C~)T(XX_).l'].JCJ. ~

x x T (l x + {U_Dxa(X_) +Dxa(X )r_}*t +

i

i=l +

!

{UxD b·i(Xx)T+D b·i(XX)UX}~Wi x x --

D b·i(XX)UxD b·i(Xx)T*t i=l x - x -

+ I{UxD c

(Xx)T+D c (XX)Ux}*p x a (l x xT x x x xT + {U D c (X ) + D c (X ) U }~ il + D c (X ) U D c (X ) ~ ii - x x x x - x -

- x

(l

(note that Dzc, which does not exist, does not appear either, because ;=0 on E!) cess

RX

of 7-5,

10-23 and 10-24)

Finally,

if we use the pro-

the explicit formula 10-22 (or rather become in this context

(again because

p=O on E):

• \7XX ( n ) T t

x

x

i f Tn _1.2 t
( 11- 3 )

(here T X is given by 6-35, with KX given by 7-2). n

§ll-b. EXISTENCE OF THE DENSITY As observed in Remark 10-33, Theorem 10-30 is weaker than Theorem 6-48.

So we will not prove 2-14 in full

generality. Rather, we will assume (A'-4) (A'-3)

instead of

(and, of course, we assume (B». is a countable union of 8 -dimensional

a

rectangles,

it is easy to find a functt'on p

a

which

meets both 10-1 and PCJ.>O on Ea' In virtue of 10-30-a,

it suffices to prove that for

MALLIA VIN'S APPROACH

149

x Ut is a.s.

invertible. Since TX is the n time of jump of one of the Poisson measures ~~ or ~, we

every tE(O,T],

have P(T~=t)=O, so it suffices to prove that U~ is a.s. x = {x x} • invertible on every set An T n _ l
so it is enough to prove that

t

+

J

TX

is a. s.

VXx(n)-l s-

n-l invertible. This being the sum of two symmetric

nonnegative matrices, it suffices that the second one be a.s.

invertible. But this immediately follows from

Lemma 7-9, and we are finished.

§ll-c.

SMOOTHNESS OF THE DENSITY

Here we assume (SB-(l;,6» and

(SC)

and at least (A'-5).

In addition we suppose that each function P a in (SB-(l;,6» meets 9-17, and we use these functions to construct the Malliavin operator. Because of (SC), a.s.,

therefore 11-3 and 7-5 yield (11-4)

where SX is given by 7-32, with k=l, k'=l. Since these ~

functions meet 7-33, we deduce that 7-34 and thus 7-50 hold. Applying 11-4 and Lemma 7-27 allows to deduce x x -i that 7-51 holds, where q (i)=E(det(U) ) (exactly as s s in 7-47). Therefore we deduce Theorems 2-27, 2-28 and 2-29 from 10-30-b,c,d respectively, with the following restrictions: we need (A'-(i+2» P~

should meet 9-17.

instead of (A'-i), and

Section 12: CONCLUDING REMARKS

1 - About the Malliavin operator on Poisson space. If the conditions 9-17 on p are not fulfilled,

in order

that the conclusion of Theorem 9-18 be true it is necessary to strengthen the hypotheses 9-16 of f and F. Then in turn,

the solution to Equation 6-2 do not bel-

ong to the domain

H~,

unless we considerably strengthen

(A'-2): for instance, sup ID c(x,.) 2 x z bounded, but also in L (E,G).

As a matter of fact, p

I

should not only be

one could replace the function

by

P

is a Cl

function on E, with values in (12 -1)

the set of

~x6

symmetric nonnegative matrices.

Then in 9-3 we should change the following:

(12-2 )

and everything else would work fine. We could even make

p depend on t. However, these generalizations do not seem to improve on the results obtained for stochastic differential equations. 2 -

The Malliavin operator is an infinitesimal genera-

tor. Likewise the proper Malliavin operator on Wiener 150

CONCLUDING REMARKS

151

space, which is the generator of an "infinite dimensional Ornstein-Uhlenbeck process" (see [19], [24]. [25]) with values in C(lO,T] ;mm) , we can interpret the operator

L

constructed in §9-a as the generator of a

Markov process M=(M)

0 as follows (rather, s> the restriction to H~ of the generator). s

M is the canonical space

The state space of

of §9-a. Knowing the initial value of

M

is as follows:

(L,H oo )

~

MO=~'

is

(n.~)

the dynamics

is a point measure whose sup-

port can be written as {(t,z(t):tED} where D is countable;

for each tED one runs a diffusion process

(Zs(t»s>O on E, with generator i

2"{ p ( z ) 6 z f + Dz P ( z) . Dz f

T

}

and starting at ZO(t)=z(t), and all these diffusion processes are independent. Then set MS is the point measure with support

Then M=(Ms)s~O is clearly Markov. Each process Z(t)

is

reversible with respect to Lebesgue measure on E, which implies that P

M

admits the canonical Poisson measure

as a stationary measure and that it is reversible

under this initial measure:

this corresponds to the

self-adjoint property for L. Also abserve that 9-17 implies that each diffusion process Z (t) lives inside E and never reaches the boundary. If we replace p by p, according to 12-1,

M

is

constructed similarly, but the generator of each Z(t) should be modified according to the first formula in

12-2. This sort of point measure-valued Markov pr9cesses

152

MALLIA VIN CALCULUS

is of course well known in other contexts: see for example Surgailis [27]. 3 -

A differential operator on

the Poisson space. When

(n,g,p) is the Wiener space of §9-b, let

tives,

be the

~m-valued func-

Hilbert space of absolutely continuous tions on [0, T]

H

with Lebesgue square-integrable deriva-

endowed with the usual scalar product. Then Shiintroduces the derivative V$ of $ER as

gekawa [23]

being the "Frechet derivative along H";

then he defi-

nes r as

and then he defines

through r (it is closely related,

L

but slightly more complicated than the approach of §9-b) . Let us come back to the Poisson space §9-a. We also have a notion of derivative cisely let ~=F(~(fl), ... ,~(fk» k

L

V~(w,s,z)

i=l

be in

V.

Then set

~

that

$(w,.)

is defined up to

~B-valued func-

tion on nx[O,T]xE. Then one can show that

which, with

More pre-

ar

a lJ(w;.)-null set. Note that V~ is an

T JV H .• s E

J

° obvious

V.

of

ax.(IJ(f l ),··,1J(f k »D z f i (s,z). (12-3)

One may prove (as in 9-4)

r (~ , '1') =

(n,~,p)

• z) V 'I' ( • , s • z)

T

r(~,~)

p (z) ~ (d s , d z)

is 02-4 )

notation, can also be written as (12-5)

(V

is not a Frechet-type derivative,

linear space;

however,

since

n

is not a

n can be viewed as an infinite-

CONCLUDING RElvIARKS

153

dimensional manifold and then one may interpret

V as a

derivative along subspaces of the tangent space).

4 -

Comparison of the two approaches. We have already

emphazised the differences a number of times, and also discussed the advantages of the first one (at least as long as smoothness problems for stochastic differential equations are concerned). The above-mentioned "derivative"

V

allows for a more thorough comparison.

In the

second approach the key role is played by (12-6)

(suppose for simplicity that there is no Wiener process and that everything is 1-dimensiona1). In the first approach we use rather

(12-7) VX

is the function on nxrO,TlxE introduced in 6-7 or 6-38. Note that in 12-7, v X does not depend on t,

where

but is predictable on nx[O,TlxE, which is not the case of VX~(w,s,z). So it seems that, mutatis mutandis,

the second ap-

proach automatically yields the "best" perturbation insuring that u~ is invertible, while in Bismut's approach we have to choose the best V X upon eXamination of the explicit formula giving DX~ (Observe also that the proof of inversibi1ity for U: is significantly easier X

than for DX t

,

in the course of proving Theorem 2-14).

REFERENCES 1. R.F. BASS, M. CRANSTON: The Malliavin calculus for pure jump processes, and applications to local time. Ann. Probab. 14,490-532,(1986). 2. K. BICHTELER: Stochastic integrators with independent increments. Zeit. fur Wahr. ~, 529-548,(i981). 3. K. BICHTELER, D. FONKEN: A simple version of the Malliavin calculus in dimension one. In Proc. Cleveland Conf. Mart. Theory. Lecture Notes in Math. 939 6-12,(1982), Springer Verlag: Berlin, Heidelberg-,-New-York. 4. K. BICHTELER, D. FONKEN: A simple version of the Malliavin calculus in dimension N. Seminar on Stoch. Processes (Evanston) 97-110,(1983). Birkhauser: Boston. 5. K. BICHTELER, J. JACOD: Calcul de Malliavin pour les diffusions avec sauts, existence d'une densite dans Ie cas uni-dimensionnel. Seminaire de Proba. XVII. Lecture Notes in Math. 986, 132-157,(1983), Springer Verlag: Berlin, Heidelberg, New-York. 6. J.M. BISMUT: Martingales, the Malliavin calculus, and hypoellipticity under general Hormander conditions. Zeit. fur Wahr. ~, 469-505,(1981). 7. J.M. BISMUT: Cal cuI des variations stochastiques et processus de sauts. Zeit. fur Wahr. ~, 147-235, (1983). 8. J.M. BISMUT: The calculus of boundary processes. Ann. Ecole Norm. Sup. ~, 507-622,(1984). 9. D. FONKEN: A simple version of Malliavin calculus with applications to the filtering theory, (1984). 10. J.B. GRAVEREAUX, J. JACOD: Operateur de Malliavin sur l'espace de Wiener-Poisson. Compte R. Acad. Sci. 300, 81-84,(1985). 11. U. HAUSSMANN: On the integral representation of Ito processes. Stochastics l, 17-27,(1979). 12. N. IKEDA, S. WATANABE: Stochastic differential equations and diffusion processes. North Holland (1979), Amsterdam. 155

156

MALLIA YIN CALCULUS

13. J. JACOD: Calcul stochastique et problemes de martingales, Lecture Notes in Math. 714 (1979), Springer Verlag: Berlin, Heidelberg, NiW=York. 14. J. JACOD: Equations differentialles lineaires, la methode de variation des constantes. Seminaire Proba. XVI, Lecture Notes in Math. 920, 442-458,(1982), Springer Verlag: Berlin, Heidelberg, New-York. 15. H. KUO: Brownian functionals and applications. Acta App!. Math. J,., 1-14,(1983). -16. S. KUSUOKA, D. STROOCK: Applications of the Malliavin calculus, Part I. Proc. 1982 Int'l Conf. Katata, Kinokuniya Publ. Co.: Tokyo. 17. R. LEANDRE: Regularite des processus de sauts degeneres, Ann. Inst. H. Poincare 21,125-146,(1985). 18. R. LEANDRE: These 3eme cycle,

Besan~on

(1984).

19. P. MALLIAVIN: Stochastic calculus of variations and hypoelliptic operators. Proc. Int'l Conf. on Stech. Diff. Equa., Kyoto 1976, 195-263. Wiley (1978): New-York. 20. P.A. MEYER: Un cours sur les integrales stochastiques, Seminaire Proba X, Lecture Notes in Math 511, Springer Verlag: Berlin, Heidelberg, New-York. - 21. J. NORRIS: Simplified Malliavin calculus. To in: Seminaire Proba. XX.

app~ar

22. H. RUBIN: Supports of convolutions of identical distributions, Proc. 5th Berkeley Symp.!I/!, 415422, (1967). Univ. Calif. Press: Berkeley. 23. I. SHIGEKAWA: Derivatives of Wiener functionals and absolute continuity of induced measures, J. Kyoto Univ. ~, 263-289,(1980). 24. D. STROOCK: The Malliavin calculus and its applications to second order parabolic differential equations, Math. Systems Theory 14, 25-65 and 141-171, (1981). 25. D. STROOCK: The Malliavin calculus and its applications. In Stochastic Integrals (D. Williams ed.), Lecture Notes in Math. 851, 394-432, (1981), Springer Verlag: Berlin, Heidelberg, New-York. 26. D. STROOCK: The Malliavin calculus, 'a functional analytic approach. J. Funct. Analysis ii, 212-258, (1981).

REFERENCES

157

27. D. SURGAILIS: On Poisson mUltiple integrals and associated equilibrium Markov processes. In Theory and Applications of random fields, Lecture Notes in Control and Inf. Sci. 49, 233-248, (1983), Springer Verlag: Berlin, Heidelberg, New-York. 28. H.G. TUCKER: Absolute continuity od infinitely divisible distributions, Pacific J. Math. 12, 11251129, (1962). 29. M. ZAKAI: The Malliavin calculus, Acta Appl. Math. (to appear).

INDEX Hypotheses (A-r) 9 (A'-r) 60.81 (A'-r) 130. 147 (B), (B') 11 (D) 22 14 (SB-(t8) (SB') 22

(SC) 14 conditions on u,v 63 73 conditions on u X • VX 14.62. 119 conditions on p. Po Terminology broad functions 13 canonical space 59 continuity (F-continuity) 34 differentiable F-differentiable 34. 50 r times differentiable 34. 74 F-(r)-differentiable 74 direct product of Malliavin operators 109 DUPG family of processes 37 extension of a Malliavin operator 104 generator 1. 2 46 graded stochastic equation grading 45 integration-by-parts setting 27 Malliavin operator 102 Peano approximation 54 Poisson space 112 Sobolev's Lemma 26 Wiener space 116

159

NOTATION 6 7 7

a, a(x), ai(x) b, b(x), b'I(X) B, B(x), WI(X)

Cr

78 83

RX

10

C~(Rn), C~(Rn), C~(Rn)

C~(q) C~,i(q)

q~(i)

7

c, c"" c', c'", Ccx

L, L' 7 10 N", N'iz

8

R 102 V,V 131

10

V*Jl, Y*!1

29

W,W~

34 37 8

det(x) OX,OXx \liX, \liX" om+n 8 xmzn 8 Ft 117

W'".x W"

67, 71 73

G,G", 6 27 G 104 H2 H~, H.., 106

H 73,82 H*W,H*t 9 27 H 7 K,K", 71,82 K" K, K", 7 102 L(<»)

6 11

62 yA. 63 SA. 61 Jl, Jl"" fl, fl", JlA. 62 6 V,V'" 27 0<1> 27 Oq> 27 Y!p y" 62 31

t.F,n,s

aG

11·IIH 1I·IIH !I1·!Il p

2

p

f( cp,'JI)

161

65 104 105 46 102

6