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)lO} 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 [tie] • Then,

the density

15

and (SB-«(;,e»

YM.-')

with e [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 80,9>0 (recall that· G (E ) a a =+00 by hypothesis. (z;;,6)-broad,

2-34 LEMMA: If Ga(C)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 all belong to . . 1 i' H.p' Applying 4-~ to ~~J=(o- ) J and to the ith compo' 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 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 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
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)



(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 )

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 o and a function

Inn a(x)l, xn

(i)

nEn z.3'

(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..

>..

(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.. >.. >.. >.. 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 . I p(z)

yj

s2q. met.

hence

a

;..

Iay-A 3 , j

i

(y,w,t)1


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

Furthermore, VkDnXx = Dnvk~ for n+k 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 1O. 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) 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 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)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 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_(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 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 tO, 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 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) p0. Since D (X )TII is locally LP bounded in X for every p0: 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) (l and

r

. c:

En

2..::p


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
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( , .• ,~ )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 in 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,


=

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 '}cR' n

n"-"II~"

n

4>' ... ' n

2

2

Thus {L'oIl~} is Cauchy in L (P'), and {4J"}c1(" with n

II¢'-'II~,

2

n

and " ... " 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(' ,")=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

).

=F(W t •..• ,W t ), where FEC (R ), 1 n p o-

= 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
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 Gin Lq(y), with g ,(s,z)=O if s
....

:= w( f n)

n

:= iiCFn)



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 pt

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 ~ 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 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 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( 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