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ANTICIPATIVE STOCHASTIC CALCULUS WITH RESPECT
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HILBERT SPACE AND TIME REVERSAL PROBLEM

presented by
Leszek Piotr Gawarecki

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_1————_—

ANTICIPATIVE STOCHASTIC CALCULUS WITH RESPECT
TO GAUSSIAN PROCESSES, STOCHASTIC KINEMATICS IN
HILBERT SPACE AND TIME REVERSAL PROBLEM

By

Leszek Piotr Gawarecki

A DISSERTATION

Submitted to
Michigan State University
in partial fulfilment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Statistics and Probability

1994

ABSTRACT

ANTICIPATIVE STOCHASTIC CALCULUS WITH RESPECT
TO GAUSSIAN PROCESSES, STOCHASTIC KINEMATICS IN
HILBERT SPACE AND TIME REVERSAL PROBLEM

By

Leszek Piotr Gawarecki

Let {Xt,t E T} be a Gaussian process with reproducing kernel Hilbert space

(RKHS) K (C) of its covariance C. We first define the Ogawa integral (5(a) of a
function u E L2((S2, J: , P); K (0)) with respect to X and prove the relation

[3(u) = 6(u) — traceDMu,

where I s is the Skorohod integral with respect to X, defined by Mandrekar and
Zhang and DM is the Malliavin derivative. For the reader’s convenience we recall
the definition and important properties of Skorohod integral. We define, in a very
general setup, the Ito-Ramer integral L, generalizing earlier work of Ramer and
Kusuoka. The integral L can be considered with respect to a Gaussian process,

under the assumption that the measure P o X ’1 on RT is Radon. We obtain that
L(u) = 6(a) — traceDFu,

where now, DF denotes the H -Fréchet derivative. This is done in Chapter 1, by
using our generalization of a result of Gross. Our work has been used to obtain

an extension of Girsanov’s theorem to the general case of Gaussian processes by

Gawarecki and Mandrekar.

In Chapter 2, we consider E. N elson’s construction of diffusion in infinite dimen-
sional case. Nelson’s work on finite dimensional diffusions has proved important
in the study of stochastic kinematics and quantum theory models. Our gener-
alization involves the choice of stochastic driving term and rigorous definition of
stochastic integral with respect to it. We explain why the stochastic driving term
for the motion is a cylindrical Brownian motion and we introduce an extension of
the stochastic integral of Metivier and Pellaumail to cover the studied case. As a
consequence we derive results parallel to those of Nelson.

In order to apply the above results to physical problems one needs to study
time reversibility of stochastic processes. A preliminary research in this direction
is presented in Chapter 3. We investigate transformations of the Skorohod integral
under maps R : T —+ T. With mild assumptions on the transformation R we

obtain that
s s X t t
1X30) (Wm) = Ix.(ut) and (DM)s m ’um) = (DMliieWRuh

where the first integral is with respect to the transformed Gaussian process and the
second is with respect to the original process X and the same refers to the Malliavin

derivative. We also investigate connections with the time reversal problem and

Skorohod-type SDE’s.

To Edyta

iv

Acknowledgment

My deepest thanks go to my thesis advisor and teacher, Professor V. Mandrekar.
He introduced me to the problems studied in the thesis and generously supported
me with his knowledge, wisdom and experience. His assistance in all aspects of
academic life was invaluable.

I appreciate the support of the Department of Statistics and Probability in
providing me with an opportunity for research and personal growth. Numerous
fruitful discussions with the faculty members will never be forgotten.

I would like to thank the members of my Guidance Committee for their com-
ments, which led to an improvement of the original presentation.

I will always be thankful to my parents for their efforts and wisdom in teaching
me what is the value of education and knowledge.

Finally, I am indebted to my wife Edyta for she supported me with all her
strength and sacrificed so much of her life. She is the most precious gift given to

me and my closest friend.

Contents

Introduction 1
1 Anticipative Integrals with respect to Gaussian Processes 4
1.1 Introduction ............................... 4
1.2 Preliminaries .............................. 5
1.3 Skorohod Integral and Stochastic Differentiation ........... 9
1.3.1 Multiple Wiener Integrals ................... 9

1.3.2 Malliavin Derivative ...................... 11

1.3.3 Skorohod Integral ........................ 13

1.4 Extension of Ogawa Integral and its Relationship to Skorohod Integral 14

1.5 Extension of Ité-Ramer integral .................... 19
1.5.1 Definition of Ité-Ramer Integral ................ 19

1.5.2 Preliminary Results ....................... 22

1.5.3 The Domain of Ito—Ramer Integral .............. 29

1.5.4 Comparison of Ito-Ramer and Skorohod Integration ..... 34

1.5.5 Ito-Ramer Integral as an Integration by Part Operator . . . 37

1.6 Examples ................................ 41

2 Kinematics of Hilbert Space Valued Stochastic Motion 48

vi

2. 1 Introduction ............................... 48
2.2 Hilbert-Schmidt and Trace Class Operators on Hilbert Space . . . . 49
2.3 Kinematics of Stochastic Motion .................... 50
2.4 Stochastic Integration in Hilbert Space ...... ' .......... 56
2.4.1 General Assumptions and their Consequences ........ 57
2.4.2 Doléans Measure of (R21) Elements of M? .......... 58
2.4.3 Inadequacy of the Isometric Stochastic Integral ....... 61
2.4.4 Cylindrical Stochastic Integration ............... 65

2.4.5 An Example Motivating Modification of the Cylindrical Stochas-
tic Integral ........................... 71

2.5 Extension of the Cylindrical Stochastic Integral and Application to

Nelson’s problem ............................ 75
3 Anticipative Stochastic Differential Equations 83
3.1 Introduction ............................... 83
3.2 Skorohod Integral under TYansformation of a Parameter Set . . . . 84
3.3 Skorohod-Type Linear Stochastic Differential Equations ....... 92
Appendix 99
A Abstract Wiener Space ......................... 99
B Backward Ito and Fisk—Stratonovich Integrals ............ 102
C Hilbert—Schmidt and Trace Class Operators on Hilbert Space . . . . 103
Bibliography 106

vii

Introduction

Anticipative Stochastic Differential Equations (SDE’s) arise in some practical prob-
lems. In the Filtering Theory, a symmetric treatment of the problem with respect
to the direction of the time flow was successfully applied by Pardoux [49]. This
technique is known as the Time Reversal of diffusion processes and it was itself of
interest of several authors: Follmer [17], Haussmann and Pardoux [23], who gave
conditions under which the time reversed process is again a diffusion and described
its infinitesimal operator. Recently, by application of Skorohod stochastic integra-
tion and Malliavin calculus, the results were improved by Millet, Nualart and Sanz
[34].

Studies of Boundary Value Problem for SDE’s lead to anticipative solutions if
the initial condition is a future dependent random variable (see Buckdahn and
Nualart [8], N ualart and Pardoux [40]). Another type of anticipative SDE’s arises
if the coefficients of the equation are allowed to be anticipative, which was studied
by Buckdahn [5H7].

Analysis of anticipative SDE’s requires extension of Girsanov Theorem. There
are two approaches to the problem. One, due to Ramer and Kusuoka, uses either
the Ramer integral (see Ramer [51], Kusuoka [29]) or the Skorohod integral (see
Nualart and Zakai [41], Buckdahn [5]-[7]). This raises the question about the re—

lationship between the Ito-Ramer and Skorohod integrals, which we study here.

Our analysis involves the Ogawa integral (see Ogawa [43]-[45]) in a natural way.
The other method is due to Bell [2]-[3] and Ustunel and Zakai [56]. It employs
Malliavin calculus and the concept of Integration by Parts Operator (IPO) with
respect to Gaussian measure, as a generalization of divergence operator. The
statement of the theorem of interest in [3] is inaccurate and we give precise gener—
alization of the divergence theorem of Goodman [20] for our setup. To carry out
this program one needs to generalize a fundamental result of Gross [22]. Then we
define the Ito-Ramer integral and we extend some work of Kusuoka [29] to the case
of Gaussian processes. The first Chapter is devoted to the above problem. This
work has been used to obtain an extension of Girsanov’s theorem to the general
case of Gaussian processes by Gawarecki and Mandrekar [18].

In the second Chapter we discuss the role of cylindrical Brownian motion in
Nelson’s Kinematic theory of stochastic motion (see [36]) in Hilbert space. First
we explain why a Hilbert space valued Brownian motion can not be recovered by
Nelson’s technique ([36]) as a stochastic driving term from a diffusion satisfying
Nelson’s regularity conditions. Therefore we consider cylindrical stochastic pro-
cesses. To construct a diffusion from Nelson’s assumptions, which is driven by a
cylindrical Brownian motion, one needs to introduce a class of integrable functions
with respect to 2—cylindrical martingales, which is larger than that used by Metivier
and Pellaumail. This requires modification of the work on stochastic integral of
Metivier and Pellaumail [35].

Applications of the results of Chapter 2 to physics require a study of Time Re-
versal problem. We begin this in Chapter 3 for Skorohod-type SDE’s. This handles
the case of reversal in time and space and, in particular, relates backward and for-
ward Brownian motion. The harder problem of determining whether time reversal

of a (non-anticipating) diffusion is again a diffusion (see Folmer [17], Haussmann

and Pardoux [23]) is presently under study but the results, being incomplete, are
not presented here. However we show that the Go and Return problem of Ogawa
[46] can be handled by our techniques. We also show some applications of our
results on transformations of Gaussian processes to line integrals of Cairoli and
Walsh [9] (see Example 3.2.1).

The Appendix contains a review of some notions used in this work. It is included

to provide the reader with an easily available reference.

Chapter 1

Anticipative Integrals with

respect to Gaussian Processes

1 . 1 Introduction

There are several goals in the development of the theory of stochastic integration.
Two of them, very natural, are enlargement of the class of integrands and enlarge-
ment of the class of integrators. We are specifically interested in a generalization
leading to anticipating integrands and Gaussian integrators.

Extension of the Ito integral to not necessarily non-anticipating integrands was
first done by Ito [25] with the help of stochastic integration with respect to quasi-
martingales. Generalization of the class of integrands to Gaussian processes was
attempted for example by Cramér [13] and Cambanis and Huang [10]. Much of
their approach was defining the integral via step functions.

We will however concentrate on different techniques. Ramer [51] introduced
a stochastic integral on an Abstract Wiener Space using functional analysis ap-

proach. He recognized this integral as an abstract version of double centered

4

stochastic integral of Ito, introduced by Shepp [54]. Ramer’s integral, further re-
ferred to as the Ito-Ramer integral, proved to be much more general object which
will be discussed later. A completely different technique, based on Wiener Chaos
Decomposition, was used by Skorohod [55] to yield an integral with respect to a
white noise random measure. This idea was further developed by Mandrekar and
Zhang [33] who obtained an integral of not necessarily non-anticipating integrands
with respect to any Gaussian process.

Another interesting attempt was made by Kuo and Russek [28], Ogawa [43],
[44], [45] and Rosinski [52], who developed a stochastic integral, also without any
special kind of measurability assumptions, with respect to a white noise random
measure on an arbitrary set. This integral was defined in terms of random series
of usual Wiener integrals. We further refer to this integral as the Ogawa integral.

In the next sections we present the above ideas with more details. We study
the relationship between the Ito-Ramer and Skorohod integrals which unify the
results on Girsanov-type theorems obtained by Ramer [51], Kusuoka [29], Nualart
and Zakai [41] and Bell [2H3]. In particular we generalize the Ito-Ramer and
Ogawa integrals (the latter appears in the course of the analysis) to the case of
an arbitrary Gaussian integrand and, by extending a result of Gross [22], we carry

out the work of Kusuoka [29] in our setup.

1 .2 Preliminaries

We begin with some selected basic concepts to make this work more self-contained.
The material concerning covariances, Reproducing Kernel Hilbert Spaces and Gaus-
sian processes was taken from the book of Billingsley [4] and papers of Chatterji

and Mandrekar [12],[30]. For further details we refer to the work of Aronszajn [1],

Cross [21] and Kuo [27].

The ideas introduced in this section lead to a useful concept of the stochastic
integral with respect to Gaussian processes defined in [21] and [31]. This stochastic
integral was used in [31] and [33] to develop the theory of stochastic integration with
respect to Gaussian processes for integrands not requiring any special measurability
assumptions.

Let T be any set and let C be a real function on T x T. C is called a covariance
on T if C(s, t) = C(t, s) and Ease,- atasC(t, s) 2 0 for all finite subsets i C T and
{0.3, s E i} C R. For a covariance C on T, there exists a unique Hilbert space H
of real valued functions on T, called the Reproducing Kernel Hilbert Space
(RKHS for short) of the covariance C, satisfying Vt E T : C ( -, t) E H and
Vt E T, f E H : (f(-),C(-,t))H = f(t). Here, for all t E T, C(-,t) denotes the

function of the first variable.

Notation. We denote a scalar product in a Hilbert space by (-, -) with possible
subscript if identification of the Hilbert space is ambiguous. For a Locally Convex
Topological Vector Space (LCTVS for short), by (~, ) we denote duality between
the space and its adjoint. Should any ambiguity arise, subscripts identifying the
space are added.

With a covariance C on T we associate a centered Gaussian process X =
{Xt, t E T} defined on a complete probability space (9, f, P), such that E (Xth) =
C (s,t), where we will always take .7: to be the a-field generated by the family
{Xt, t E T}. Without loss of generality we assume that all probability spaces con-
sidered here are complete. Denote by H (X) the closed linear span of {Xt, t E T}
in L2(Q,.7-', P). Note that if Y1,Y2, ...,Yn E H(X) then (Y1,Y2,...,Yn) is a multi-
variate normal variable. Then the RKHS H of the process X is of the following

form:

H = {f : f(t) = E(XtYf), for a unique Yf E H(X)}

Let 7r : H —-> H (X) be a map defined by: 7r( f) = Yf. Then 7r is an isometry.
In particular 7r(C(-, t)) = Xt.

Definition 1.2.1 (1} The isometry 7r : H —> H (X) is called a stochastic inte-
gral with respect to Gaussian processX .
(2) If K is a Hilbert space isometric to the RKHS H of the Gaussian process X

under an isometry V then we define a stochastic integral S of any k E K with
respect to X by S(k) = 7r(V(k)).

Several interesting examples of Gaussian processes, their RKHS’s and stochastic
integrals with respect to these processes can be found in [30] and [12]. We present

here only those examples which we discuss later.
Example 1.2.1 Gaussian processes.

(a) Brownian motion. Let T = {(t1,...,tn) = t E R“, t,- Z O} and define
C(t, t’) = f=1(t,- /\ t;). Then the function C is a covariance on T . For n = 1, the
associated Gaussian process is called the Wiener-Lévy Brownian motion or
Brownian motion for short, and for n > 1 the associated Gaussian process is

called the Cameron-Yeh process.

The RKHS H of the covariance C is given by

H = {f: N) = jot" [0 t1 9a., ...,u.)du....du.., g e 12.01“»

with the scalar product

(M2) = /0°° /O°°g.<u>gz<u>du

7

where du is Lebesgue measure on R”.

Denote the Brownian motion process by B and consider stochastic integral with
respect to this process. For h E H, 7r(h) = fol h’ dB, h’ denoting the derivative of
h and the last integral is the Wiener integral. Indeed, it is an isometry between
H and H (B ) because the Wiener integral is an isometry between L2([0,1]) and
H(B) ([27]). Also 7r(C(-,t)) = B, = fol 1[0,t](s)st, 1A(-) being the characteristic
function of set A. The RKHS H is isometric to the Hilbert space L2([0,1]), with
the Borel a-field and Lebesgue measure, by an isometry V( f )(t) = f; f (s)ds, for
f E L2([0,1]), t 6 [0,1]. The stochastic integral S for functions from L2([0,1]) is
then just the Wiener integral, S(f) = 7r(V(f)) = fol de.

(b) Gaussian white noise measure. Let (5,2, ,a) be a a-finite measurable
space, T = {A E Z : a(A) < oo} . For A,A’ E 2 define C(A, A’) = a(AflA’).
Then the function C is a covariance on T.The associated Gaussian process is called

Gaussian white noise measure. The RKHS of covariance C is given by

H = {f = f(A) = [A f(U)u(dU). f e L2(S.E,u)}-

Now let us consider stochastic integral with respect to Gaussian white noise
measure. The map V: L2(S,E,u) —+ H given by V(lA)(-) = a(- n A) = C(-,A)
is an isometry. Then the stochastic integral S : L2(S, 2, p.) —) H (X) is defined
by S(f) = 7r(V(f)). In case of S = [O,1],Z - the Borel a-field and ,u — Lebesgue
measure, the stochastic integral S is the Wiener integral.

(0) Generalized Gaussian process. Let T = C8°(G'), the space of smooth
(i.e. infinitely differentiable) functions with compact support in a bounded do-
main G with a smooth boundary in R". For qb 6 08° (C), we denote (D“¢)(a:) =
alal¢(ar)/893‘f1...3xg“, where la] = 23;, ca, oz,- are non-negative integers. If C is

a covariance on T, then the associated Gaussian process is called a generalized

Gaussian process.

(c1) In the case of
Com.) = [G ¢1(U)¢2(U)u(dU),

where a is Lebesgue measure on R", the associated process is called Gaussian

white noise. The RKHS of C is L2(G, a(du)).

(c2) For
C(¢1,¢2)=Z/G((D°‘¢1u(I)“¢2(U))/i(dU),

|oz|_<_m

the associated Gaussian process is called Gaussian White noise of order m.

The RKHS of C is the Sobolev space H5”(G).

1.3 Skorohod Integral and Stochastic Differen-
tiation

The Skorohod integral and stochastic differentiation as presented here was intro-
duced by Mandrekar and Zhang and most of this section recalls results of [33]. For
the original work of Skorohod we refer to [55].

1.3.1 Multiple Wiener Integrals

For the detailed construction of Multiple Wiener Integrals with respect to Gaussian
processes we refer to [33]. For the original construction of Ito see [24].
Let C be a covariance defined on an arbitrary set T with the RKHS H and let

p be a non-negative integer. Tensor product H ‘8’” of p copies of RKHS’s H consists

ll

of all functions of p variables of the following form:

f(tla t2a "-7 tp) : Z aal,a2,...,apea1(tl)ea2(t2)---eap(tp)

a1,a2,...,ap

with Za1.a2....,ap agha2map < 00. Here {em 01 = 1, 2, ...} is an ONB of H and the
summation is over all tuples (a1, ..., 0110). Furthermore, the scalar product of two

functions f, g E H 8’? is defined as

(fa g)H®P : Z aal,a2,...,apba1,a2 ..... 01p)
al,a2,...,ap
if
f(tlatZa "-atp) : Z aa1,a2,...,apeal (t1)ea2 (t2)---eap (tp)7
a1,a2,...,ap
g(t1, t2, ..., tp) = Z b0,1m ,,,,, 0,1,60,1(t1)eo,2 (t2)...eap (tp).

a1,a2,.u,ap

Notation. For f E H 8’? we denote by f its symmetrization, which is defined as
f = #23 f (ts(1)a ts(2)a ..., ts(p)), where the sum is over all permutations s of the set
{1, ..., p}. We denote by H 0” the pth symmetrized tensor product of H, which is
a Hilbert subspace of H 8’1” consisting of all symmetric functions in t1, ..., tn. Also,
H690 2 HC90 = R (real numbers). Note that if f E H699 then f 6 H91”.

For any p = 0,1,..., Multiple Wiener integral, 1,, is a linear map from H W
to L2(Q,.7:, P), where (9,37, P) is the underlying probability space on which the
Gaussian process X is defined. The integral is determined by the following prop-
erties:

(1>I o(f)=fforf€H®°=R-

(2)11(f)= 7r(f) for f E H®1=H

<3) 1p+1(fg)— — I mug) — Lap—logy) for f 6 He”, 9 6 He and f<§9 =
(f(t1,.. ..t p) 9(tk))H'

(4) [le (f :llL2(fl): pillfilH®p for fp E Hm-

10

Below we list some other useful properties of Multiple Wiener Integral. For
f,g E H®p and h 6 H8”, we have,
(5)Ip(f f)=Ip(f~)'
(6 ) E{I (m— — o and E{I()1p(g)} =p!<f, mm
(7)1 p(HQZ’l’) 110(9) Ik(H®k) for all k < p.
(8) Ip+1(fh)— - I (f)11(h) —— 53:1 Ip—1(f§h) for f e W, h e W and fgh =
(f(t1,--- t p)ah(tk))H-

Let f E kH®p, f(t1,t2, ...,tp) : ea,(t1)eaz(t2)...eap(tp) where among
a1,...,ap only n are different with repeats p1,p2,...,p,,, p1 +p2 + + pa = p.
Denote the corresponding n different eai’s as u1,u2,...,un and assume that they
are orthonormal elements of H. Then Ip( f) = n _1 ’HP,(I1(u,-)) where ’Hp,’s are
Hermite polynomials normed in the following way: Hp(t) = 7i2?HP(7t'2') with
Hp(t) = (—1)Pe‘”2 $642.

A version of Wiener Chaos Decomposition is given in the next Lemma.

Lemma 1.3.1 L2(§2,}',P)= EBp—ol p(HQP).

1.3.2 Malliavin Derivative

The Malliavin derivative defined in this section plays an analogous role in Skorohod
stochastic calculus with respect to Gaussian processes as the Malliavin derivative
in case of Skorohod calculus with respect to Brownian motion.

Let u = {uh t E T} be a measurable stochastic process defined on a probability
space ((2, .7, P), such that u.(w) can be considered as an H -valued random variable
in Bochner sense. We assume that u.(w) 6 L262, H), in particular, E Hung, < 00.

This condition implies that at E L2(Q) for each t E T in view of the following

11

inequality:
uf(W) = (U-(w),C(nt))ii _<_ Ila-(W)II§;C(t,t)-

By Lemma 1.3.1, for each t E T, there exist unique f;(-) 6 H91”, p = 0,1,...,
f;(t1, Mat?) 2 fp(t1a ...,tp,t) E H®(P+1), SUCh that

ut<w> = i5 1mg) = i1p(fp<t.,...,tp,t>>. (1.1)

p=0 p=0

For computation of the L2(Q, H) norm of a stochastic process u it is very useful
that for ut = Ip(f(-,t)) and vt = [g(g(-,t)), where f(-,-) E H®<p+1), g(-,-) E
H®<q+1> and for each fixed t E T, f(-,t) 6 He” and g(-,t) 6 H99, we have,

E{(U.,'U.)H} : { p!(fag)H®(p+1) ifp : q

0 fip#q'

Now we recall definition of Malliavin derivative.

Definition 1.3.1 Let u E L2(Q,H). By the Malliavin derivative Dfi’jut for
fixed h E H we understand a random variable in L2(Q), defined as a limit of the

following series:

imp—«(flab ...,tp_1,s,t>,h<s>)>

p:

where ut has the unique representation (1.1). If for fixed t E T, the Malliavin
derivative DQ’Iut erists for all h E H and the series

(X3

pIp—1(fp(t17 ...,tp_1,8,t))
p=1

defines a random variable in L2(SZ,H), then we define the Malliavin deriva-
tive Dyut E H, as a function of argument 3, in the following way: D3421, =

23:1 pI _1(fp(t1, ...,tp_1, s,t)). In this case (Dyut, h(s))H = D,1,”ut.

12

We give sufficient and necessary conditions for existence of Malliavin derivative

as well as for some regularity of this derivative in the following Lemma.

Lemma 1.3.2 Let u 6 L262, H) where ut has the representation (1.1).
(I) Let t E T be fired. 193%, 6 L262, H) exists if?”

00
prlllfphtfllfisp < oo
p=1

and in this case

Dig/fut : ZpIp—l(fp('asit)) and END-Mitt“; = prillfp('at)i|%{®l’ < 00'

(2) The Malliavin derivative Dsut E L2(§Z, H ‘32), i.e. it is a Hilbert-Schmidt oper-
ator, iff

00
219p! “fplli{®(p+1) < 00.
P=1

Example 1.3.1 Malliavin derivative for Brownian motion.

In the case of standard Brownian motion, Multiple Wiener Integrals 1,, and con—
sequently, the Malliavin derivative defined above, coincide with Multiple Wiener

Integrals I; and the Malliavin derivative Di defined in [41]. More precisely,
[g(fp) = [pa/W f) and DSF = V(DiF)(s)

for any fp E L2([O,1]P) and F E L2(SZ) with the first equality in L262) and the
second in L262, H). Here V : L2([O,1]) ——> H is defined by: Vf 2 f0' f(s)ds (clearly
V®pr e H®P and VDI‘F e L2(o, H)).

1 .3.3 Skorohod Integral

We are ready to recall definition of the Skorohod integral, which is based on the

Wiener Chaos Decomposition of Lemma 1.3.1.

13

Definition 1.3.2 Let u E L262, H) has the decomposition (1.1). If

2 Ip+1(fp) = Z Ip+1(fp) converges in L2 (9)

p=0 p=0
then the sum is called the Skorohod integral of u and is denoted by I 3 (u).

Note that for u 6 L262, H), we have, I8(u) E L2(SZ) ifi 21:1(p+1)!||fp||§{®(p+1) is
finite and in this case, the L262) norm of the Skorohod integral I 3 (u) coincides with
the above sum. Furthermore, the domain of the Skorohod integral I 3 consists of
all u 6 L262, H) for which the above sum is finite. As we can see the measurability
condition for the integrand in the Ito integral is replaced in the Skorohod integral

by a ”growth” condition.
Example 1.3.2 Skorohod integral with respect to Brownian motion.
As a continuation of Example 1.3.1, we have
Ii(u) = [S(Vu)

for u E L2(Q,L2([0,1])) (clearly Vu E L2(§2,H)). Here Ii(u) is the Skorohod
integral defined in [41]. In the case when u is adapted to the natural filtration
of Brownian motion, ft = 0{B3,s g t}, then the Skorohod and Ito integrals
coincide: I ’(Vu) = I 2(u) = fol utdBt. If u is adapted to the future filtration
.77‘ = o{81 — B3,t S s g 1} then the Skorohod and backward Ito integrals coincide
(see [41]).

1.4 Extension of Ogawa Integral and its Rela-

tionship to Skorohod Integral

In this section we introduce the Ogawa integral with respect to Gaussian process

X = {Xt, t E T} defined on a probability space (SLIP, P). For original definition

14

and for properties of the Ogawa integral we refer to [43]-[45],[28],[52],[41].

Definition 1.4.1 Let u = {uht E T} be a stochastic process on (9,]: , P), such
that u is an H -valued random variable. Let {en}$,;1 C H be an ONB in H. Assume
that H(Humq < oo) = 1.
(1) A process u is called Ogawa integrable with respect to the process X
and ONB {an}?=1 C H if the following series converges in probability:

62(u) = 2(u, en)H7r(en).

n=1

In this case 62(u) is called the Ogawa integral of the process u with respect to X
and ONB {en},‘.’,°=1.
{2) If the limit in (1) exists with respect to all ONB’s of H and does not depend on

the choice of basis, then process u is called universally Ogawa integrable with

respect to X and 6°(u) denotes its Ogawa integral.

To obtain the relationship between the Ogawa and Skorohod integrals one only

needs the following technical Lemma, which is an analogue of Proposition 3.5 [41].

Lemma 1.4.1 Let F E L2(Q) be such that its Malliavin derivative
DMF E L2(Q,H) and let f E H. Then,

13(Ff) = I$(f)F— (DMFJCD- (1-2)
Proof. Let F = Im(fm). Then,

13(Ff) = Im+1(f'mf)
= Im(fm)I1(f) — mIm_1((fm(t1,---,tm—1,')af()l)
= Fro) —D?‘F-

15

If F: Enrol m(fm), then Ff E D(IS) because

00

Z(m + 1)! II—{fm(t1, tm)f(t)

m=0
m
+me( tla-"t i- 1)t ti+1w~ tm)f(ti)}”i{®(m+1)

=1
00

m!
3 Z In—+1( m + 1)2 “fmfllimonm
m=0
=2 m'( (m + 1) )llfmllyamllflly < 00
m=1

since E(||DMF||}°2,) 2 23:0 m!m||fm||§1,®m < 00. Because
1300;" Fr) = E(|(D.MF, f(cc))l2) s EIIDMFlliillflliq < 00.

we have

2 mIm— 1 ((fm(t1a---at m— 1") f())) _>D_1wa

m=1
in L262) as N —> 00 and therefore 2N_11m(fm)11(f) converges in L262) to
15(Ff)— DMF and (1.2) is valid for F=Zm_01m(fm), i.e. for any F 6 L262).

[:1

Proposition 1.4.1 Let u E L2(Q,H) and assume that the Malliavin derivative
DMu(w), of u exists and for every w E (I it is a Hilbert-Schmidt operator on H,
with E IIDM Ulliigz < oo. Assume furthermore that DMu(w) is even a trace class

operator on H for every w E St. Then,
u 6 19(13) 0 13(6") and 6°(u) = Is(u) + trDMu u a.e.

Proof. The statement, u 6 13(13), follows from Theorem 3.1 in [33]. Let
PN E ’P(H), PN 2: 25:1 hk <8) hk where {hk}‘,:°=1 is an ONB in H. Compute the

16

expression for I S(PNu). Begin with

13((Uahk)hk) = (“$013010—(DM((u,hk))ahk)
= (u. hk)Is(hk) — (1721“, his)
= (u, karat.) — (DMu(hk), hk).

The first equality is a consequence of Lemma 1.4.1. The second can be justified

as follows. Let u — —Zm_OI m(fm) be Wiener chaos expansion of u with fm E
H®(m+1),symmetric with respect to the first In variables, as proved to be possible
in Lemma 3.1 [33]. Since u is given by an L262, H) convergent series Zm-01(fm)1

we have,

i Im((fmahk)) : (Zn: Im(fm)ahk) —> (u, hk)

m=0 m=0

in L262). This means that (u, hk) has the following series representation:

(Milk) = Z Im((fmah'k))
m=0
with (fm, hk) E H 9m. Consequently,
WWW.) = (if mam... hum.)
m=1
= (Dfiufik)

= (DMU, hk ® hk)

Finally, we get,

[3 ()(PNuw )=(u(w§: MfiM (1))hk,hk) (1.3)

k=1 k=1
i.e.

[3(PNu) = 6°(PNu) — tr(PNDMu).

17

Next we want to show that I S(PNu) —> 13(u) in L2(Q). It is enough to prove
that
E II PNu—ulfii —) 0
E || DMPNu — DMu ”1,38,. —> o
as N —> 00 (see Theorem 3.1 in [33]).
Because PN —> I H strongly (1;; denotes the identity operator on H),

we have,

M PNu(w) — u(w) ||H—> 0 Vw E 9.

Therefore E || PNu — u I|§,—> O by the Lebesgue dominated convergence theorem.
Also,

N N
DMPNu = DM(§:(u, huh.) = Z DM(u, hk)hk
16:1 16:1
implies
00 N
H DMPN“ — DMU Him = Z{Z(DM(’U: hk), hj)(hka hj) — (DMu(hj)ahj)}2
j=1 k=1
= Z (DMu(h,-),h,-)2 —> 0
j=N+1
as N —> 00, because || DMu ||H®2< 00.
Since
N
H DMPNUI) Him: Z(DMU(hg-)ahj)2 SH DMU Him,
j=1

we obtain, again by the Lebesgue dominated convergence theorem, that
E || DMPNu — DMu “Ema 0.

This proves that I S(PNu) —> 13(u) in L262). Since trPNDMu(w) -—) trDMu(w)
for every w E 0 we have 25:1(u, hk)7r(hk)(w) converges in probability, indepen-

dently of the choice of the ONB in H.

18

...I

1.5 Extension of Ito—Ramer integral

In this section we extend the Ito-Ramer integral and give some properties of this
extended integral, which are parallel to those stated in Ramer [51] and Kusuoka
[29]. In these papers the Ito-Ramer integral was defined in two slightly different
ways. The main objective of both authors was to give a solution of the problem
of absolute continuity of non-linear transformations of a Gaussian measure on a
Banach space, which was first considered by Cameron and Martin [11]. Our work
on the Ito-Ramer integral is inspired by the ideas developed in [51] and [29].

Let (i, H, B) be an abstract Wiener space ([21]) and u be standard Wiener
measure on B i.e. the measure induced by isonormal cylindrical measure on H by
i. Ramer and Kusuoka considered a transformation T = I + F, where F : B -—> H
was such that DF, the Gateaux derivative of F in the direction of H, existed and
for each x E B, DF (x) was a Hilbert-Schmidt operator on H. Then under certain
conditions on T and F, the authors showed that,

d(II0T)

l
d (x) = dC(IH + DK(x)) exp{—” < Kx,x > —trDK(x)” — 5 |Kx|i1}.
u

Here, ”< K x,x > — tr DK(x)” was called the Ité-Ramer Integral. We extend

this integral for a very general setup as follows.

1.5.1 Definition of Ito-Ramer Integral

Let {Xt, ,t E T} be a Gaussian process defined on a probability space (9, f, P).
We consider Kolmogorov functional representation of the process X, i.e. the prob—
ability space RT, with the o-field RT generated by cylinder sets, and probability
measure ,u, such that the finite dimensional distributions of the canonical process

x(t) 6 RT coincide with the finite dimensional distributions of the process X. RT

19

becomes a LCTVS when equipped with the product (Tihonov) topology. We as-
sume that the measure It on (RT, RT) is Radon and we denote its support by X
(see Proposition 3.4 in [57] and for an example of non-existence of a support in
case of a non-Radon measure see [14]). Then H, the RKHS of X is separable. The
measure [1 is Gaussian and X = HRT C RT (the closure of H in the topology of
RT). The triple (i, H, X) is not necessarily an Abstract Wiener Space (AWS, see
Appendix) but the following relation holds:

X*‘-*>H‘->X
i i
where 2* is the conjugate map to 2. Both 2* and i are continuous, dense embeddings.
Example 1.5.1 Stochastic integral of a linear functional.

The stochastic integral, introduced in Definition 1.2.1, of e E X“ is given by
7r(e)(x) = e(x) a.e. [x(dx).

We consider atriple (i, H, Z) where (i, H, Z) = (i, H, B) is an AWS or (i, H, Z) =
(i, H, X) is the triple associated with some Gaussian process. Let E be a real Ba-

nach space and L(H, E) denote the space of bounded linear operators from H
to E.

Definition 1.5.1 {1 ) A map f : R —> E is called absolutely continuous if for

any —00 < a < b < 00 and e > 0, there exists some (5(5, a, b) > 0 such that
f=1||f(t,) — f(s,)||E < 5 holds for any integer n and a S t1 < 81 S t2 < 52...t,, <

8,, _<_ b, Z?=1|t,-— si| < 5(e,a, b).

(2) A map f : R —) E is called strictly absolutely continuous if it is continu-

ous, strongly difierentiable almost everywhere and it satisfies that f: H (df/dt) (t) H Edt

2O

is finite and f(b)—f(a) : f:(df/dt) (t)dt for any —00 < a, b < oo , where (df/dt)(t)

denotes strong derivative of f at t.

We note that given a map f : R —> E, f is absolutely continuous if it is strictly
absolutely continuous. In the case of reflexive Banach space E absolute continuity

of f : R —> E implies its strict absolute continuity.

Definition 1.5.2 A strongly measurable map ( in the sense of Bochner) F : Z —> E
is said to be Stochastic Gateaux H-Differentiable (SOD) if there exists a
strongly measurable map DGF : Z —> L(H, E) such that

is), F(2 + th) — F<z>> —> to. DGF<z>h>

in probability u as t —> O for every (p E E* and h E H. DGF is called the

Stochastic Gateaux H-Derivative of F.

Definition 1.5.3 A strongly measurable map F : Z —-) E is called Ray Abso-
lutely Continuous (RA C) if for every h E H there exists a strongly measurable
map Fh : Z —> E such that u(Fh = F) = 1 and Fh(z + th) is strictly absolutely

continuous in t for each z E Z.

Definition 1.5.4 A map F : Z —> E belongs to class H1(Z —> E;du) ifF is SGD
and RA 0.

Notation. For K, a linear subspace of H, we denote by P(K) the set of all finite

dimensional projections of H with range in K.

Now we define the Ito-Ramer integral with respect to a Gaussian process X.

21

Definition 1.5.5 A map F : Z -—> H is said to belong to D(L), the domain of the
Ité-Ramer integral, if the following conditions are satisfied .'

(1) F E H1(Z -—> H;du).

(2) DGF(z) E H®2 u a.e.

(3) there exists a measurable function LF : Z ——> R such that

LpF(z) 2: (PF(z), z) — trPDGF(z) —> LF(z)
in probability p. as P ——> IdH, P E P(Z*).

Remark 1.5.1 (1) In the definition of the Ito-Ramer integral we consider only fi—
nite dimensional projections P E P(Z*). If the triple (i, H, Z) is an AWS (i, H, B)
then the above definition coincides with the definition of Ito-Ramer integral given
in [29] for projections in ’P(H) with ranges in B“.

(2) If assumption (1) in Definition 1.5.5 is replaced by the requirement that F be
continuously Gateaux H -difierentiable, then Definition 1.5.5 coincides with the one

given in Lemma 4.2 [51].

From now on we will concentrate on the case when (i, H, Z) = (i, H, X) is the
triple associated with Gaussian process X. We will return to the case (i, H, Z) =

(i, H, B) in examples on Brownian motion.

1.5.2 Preliminary Results

In order to study the domain of the Ito-Ramer integral with respect to Gaussian
processes we need some general results. We begin with an extension of Fubini-type

theorem (Remark 2.2 in Gross [22]). The result in [22] is justified with help of AWS

arguments. Our reasoning is based on Karhiinen-Loéve representation ([33]).

22

Proposition 1.5.1 Let (i, H, X) be as in Section 1.5.1 and u be a Gaussian mea-
sure supported by X. Let K Q X be a finite-dimensional linear subspace with K g
X“ and {k1, k2, . . . , kn} C K, its orthonormal basis (ONB). Let L = 03.;1 ker(kj)
( a closed complement of K in X j and denote by PK and PL, the projections of X
onto K and L resp. Define MK = Bat, 11;, 2 Put, the image measures under PK

and PL. Then we have the following equalities:

[X f($)#(d$) = [UK f(a: + i)uL(dx) ® ,u.K(dx) (1.4)
2 Lan f (x +ngkj) ”L(dml ‘8 (21;)? exp (—%j2:x§) dx1 . . . dxm

for any measurable function f : X —) R+.

Remark 1.5.2 The formulation of the above proposition is correct. X is a Haus-
dorfir LCTVS and for any Hausdorfi LCTVS if K is its finite dimensional subspace,
then K is closed and L C X defined as above is its closed complement, LEBK = X.
If x E X then x can be decomposed in a unique way into x = xL +xK with x1, E L

and xK E K. Projections PL and PK are linear and continuous (in our case

PK(x) = xK 2 31:1 kj(x)kj and PL(x) = x — xK).

Proof. (of the Proposition) Because PL,PK are linear and continuous, the
image measures PLu,PK/1 are Gaussian measures on L and K respectively. We
want to prove that u = uL®uK on L x K = X. First we will prove that uL®uK is a
Gaussian measure and then, that functionals on X can be decomposed into a sum
of two independent (with respect to the measure u) Gaussian random variables
related to subspaces L and K.

Claim 1. [IL (8) #K is a Gaussian measure.

23

Proof(of Claim 1) ch E X *, (p 0 PL, (,0 0 PK are independent Gaussian random

variables with respect to I11, (8) fix on X. This is because

90 0 PL(37) = $00 PL(~’13L +51%) 2 99(le

hence, (,0 o PLIL = cplL E L* is normally distributed with respect to #1,.

Thus go 0 PL on X is also normally distributed with respect to I11, (8) pg since values
of this functional are independent of the component belonging to K. By the same
argument (p 0 PK and (4p 0 PL, (,0 0 PK) are Gaussian. Independence follows from

the equalities below,

[1: 90 0 PL(~T)90 0 PK($)#L ‘8 ”x(dflil
= foK 90 0 PL($L + 33K)<P 0 PK(IBL + $K)d#L ® MK
2 / SOIL($L)<P]K(CBK)dHL (X) #K
LxK
= /r|L(xL)duL/ r|K(:vK)dIIK
L K

= fxsoomvdm.aux/XsooPKWuLsux

Finally 90 _—_. (p 0 PL + (,0 0 PK is a Gaussian random variable with respect to
,aL (8) fig on X. This completes the proof of Claim 1.
Claim 2. (p 0 PL, (,0 0 PK are independent Gaussian random variables relative to ,u
on X.

Proof(of Claim 2). We E X*, go(x) = Zfi1(<,o,e,)7r(e,-)(x) = 21(go,e,-)e,-(x),
for ONB {6,},921 in H, where {6,},921 C X* and e,- = 19,-, (i = 1,...,n) (see [33]).
Therefore we can express compositions of functional (,0 with projections PK and

PL as follows:

roPK(w) = Z(r.ei)ei(m)

24

n

IpoPL(x) = :(fpaei)ei($— ej(5’7)€j)

j=1
= 20109506433) —§:(<P,€i)€i($) = Z: (90,e,-)e,-(x)
i=1 i=1 i=n+1
Because {ed-'5) = 7r(e,-)(x) 3:1 and {61(1B)},9°:n+1 are independent families of

random variables with respect to u, also (,0 0 PL and (p 0 PK are independent. Claim
2 is proved.

Now to prove that ,u :2 ML (8) ,uK we compare characteristic functionals of these

measures.

ut@7ux(<e) = fxexpfiflfllméwfldr)
= I. expvoo e(x) +900 mom. swam)
= fexp1leL(x.>}uL<de> [K explir|x(xx)}ux(dmx)
= f, GXPIW o mamas) f, expw o manage)

= f, exp{ir($)}u(dx) = no).
Because u 2 ML ® uK, we get,

I. f(x)u(dw) = foKfWL MK»... e Mombasa.)

for any measurable function f : X —+ R+. Now, L 2 39:1 kerkj, therefore the
random vector (k1, ..., kn) has the same distribution under both measures MK and

u = m, ®uK, that is n-dimensional standard normal. Hence equation (1.4) follows.

El

Next we extend the results of Kusuoka, contained in paragraph 4 of [29], that
are relevant to our work. Kusuoka was concerned the setup of AWS while we
are interested in a more general situation of the triple (i, H, X) associated with a

Gaussian process.

25

Definition 1.5.6 Let A C X be any subset. Define function
p(' ;A) = X —> [0,+0<>l by

+oo otherwise

inf{||h]]H : 32+}. 6 A} if(A—x) nH ,I i
p($;A)=

Next Proposition can be proved in the same way as Proposition 4.1 in [29].

Proposition 1.5.2 (1) If subsets A and A’ of X satisfy A C A’ then p(x;A) Z
p(x;A’) Vx E X.

(2) VA C X, h E H, x E X , p(x+h;A) S [Ihl|H+p(x;A).

{3) Let {A,,}f,°=1 be an increasing sequence of subsets of X and A = U31, An, then
Vx E X p(x; An) \, p(x;A) as n —+ 00.

Theorem 1.5.1 (1) IfK is a compact subset of X, then p(-;K) : X —> [0, +00] is
lower semi-continuous.

(2) If G is a o-compact subset of X, then p(-; G) : X —> [0,+oo] is measurable.

Proof. Since (2) is a consequence of (1) and Proposition 1.5.2 (3), it is enough
to prove (1). We follow the idea of proof given in [29].

Define A0 = {x E X : p(x, K) g a}, B (a) - the closed ball of radius a, centered
at O, in H. We want to show that A, =- K + B (a). The inclusion AG 3 K + B (a) is
clear. For the opposite inclusion, take x E Aa. Then 3{h,,}f,°=1 C (K — x) flH such
that ”hn“ S a + %. Being norm bounded, the sequence {h,,},°,°=1 contains a weakly
convergent subsequence {hnk}z‘_’__1. Let h E H denotes its limit. Since X * C H and
Vt E T xt(h) = h(t) (point evaluation) is an element of X * we also have hm, —> h

in X (convergence in X is a pointwise convergence). Also,

llhllH : SUP{|(h,CE)| W E X*, ”33“}; S 1}

26

: Sllp{kli)ncl)lo](hnk,$)l ;CC 6 Xi!) llxllH S 1}

s 113...... ”an. 3a.

Thus h E B(a). Since K C X is compact and hnk —> h in X with x + hm. E K,

(k = 1,2, ...), also x + h E K and therefore x E K + B(a). Thus A = K + B(a).
We claim that B (a) C X is closed. Indeed, we have the following:

Lemma. Let X be a reflexive Banach space and Y be a LCTVS.

Let T : X —> Y be linear and continuous. Then T(BX(0,1)) C Y is closed, where
B X (0, 1) is a closed unit ball centered at 0 in X.

Proof(of Lemma). T : X —> Y is linear and continuous, hence T : Xw —> Yw is
linear and continuous (w - means weak topology). This is because if {xa} is a net
in X with xa —> x in Xw, then Vy“ E Y*, y*(Txa) = (y*T)xa ——> (y*T)x = y*(Tx),
for (y*T) E X“.

Because X ** E X by the canonical isomorphism K3, we get,T o n—1 : X:,* —+ Yw
is linear and continuous and further, T 0 K71 : X51, —> Y... is linear and continuous
(where w — >1: denotes the w — * topology). The latter holds because reflexivity of
X implies reflexivity of X *. Now, the closed unit ball B X... (0, 1) is w — * compact
by Alaoglu-Banach theorem. That means K.(Bx(0,1)) is w — * compact in X **,
hence To n“1(n(BX(O, 1)) = T(BX(0,1)) is to closed in Y. Because Y and Yw have
the same closed, convex sets, T(BX(O,1)) is closed in the topology of Y and the
lemma is proved.

Thus i(B(a)) C X is closed, therefore A0 = K + B (a) C X is closed.

Next theorem can be proved as in [29] with obvious modifications.

Theorem 1.5.2 Let E be a separable, reflexive Banach space and F : X —-> E

be a measurable map and suppose that there exists a constant c > 0 such that

27

Vx E X,h E H, “F(x + h) — F(x)||E _<_ c||h||H. Then there exists a measurable
subset D0 ofX and a map DF : X —+ L(H, E) such that :

(1) #(Do) = 1-

(2) limHO -:—(F(x + th) — F(x)) 2 DF(x)h, Vx E D0, h E H.

(3) DF(-)h : X —> E is measurable Vh E H.

In particular, if DF : X —% L(H, E) is strongly measurable then

F E H1(X —§ E;du).

Corollary 1.5.1 Let G be a o-compact subset of X and gt be a smooth function
with compact support in R. Then g() = ¢(p(- ; G)) : X —+ R ,with the convention
that 45(00) 2 0, belongs to H1(X —> R; d,a) and

61¢
IIDG9(~”3)HH S SWINE] 3 t E R}
for u a.e. x.

Proof. First we observe that by Theorem 1.5.1, (2) g is measurable. Also,
d¢
dt

d¢
a“

”9(33 + h) - g(cv)” S sup{ (t) ;t E R} (p(a? + ’74 G) - p(x, G))

 

 

S sup{

v

;t E R} llhllH

by Proposition 1.5.2, (2), (with the convention oo — oo = 0, note that p(x +
h; G) = 00 iff p(x; G) z 00). Therefore, assumptions of Theorem 1.5.2 are satisfied.
DGg(-)h can be thought ofas h(DGg(-)) with h E H“, DGg(-) : X —> H“. Thus we
have DGg : X —) H * is weakly measurable (by (3) of Theorem 1.5.2) and therefore
it is strongly measurable in view of separability of H. The inequality at the end

of Corollary is obvious.

28

 

1.5.3 The Domain of Ito-Ramer Integral

In this section we specify two subclasses of the domain of the It6—Ramer integral.
We begin with generalization of Ramer’s result, Lemma 4.2 in [51]. As a tool we
use the inequality given by Ramer in Lemma 4.1, [51] but we need it for functions

in H1(R” —> R”, 7”), where 7,, is the standard Gauss measure on R” (see [29]).

Note. By H m we denote the tensor product H (8) H which is identified with the

Hilbert space of Hilbert-Schmidt operators on H.

Lemma 1.5.1 Let f : R" —> R" be an H1(R"’ ——> R",’y,,) function. Assume ||f||Rn
and ”Ba f ”(Rn)®2 E L2(R",'y,,) (the latter norm is the Hilbert-Schmidt norm).
Then,

/ "(ovum — tTDGf($))27n(d$) s Lamont. + llDGf(w)||?Rn)®2)7n(drv).

Theorem 1.5.3 Let F E H1(X ——> H;du) and assume that DGF(x) E H692 for ,u
a.e. x and F E L2(X,H), DGF E L2(X,H®2). Then F E D(L) and

/, ILF(a:)l2II(dI:) s Lamont. + HDGF<x>IIi®2>u<dsc>.

Proof. We use Proposition 1.5.1 to extend Lemma 4.2 in [51] to our case.
Any Pn E P(X*) with dim Pn(H) = n can be written as follows:
Pn226i®6i, €i€X*, {83' 21:1 ONB in H.
i=1

We will first show that {L pnF }f,°=1 is a Cauchy sequence in L2(X).

/,,<LP.F(:c> — mer<x>>2u<dm>

= lam — mm»). 32> — ma — pm)DcF(.)}2,.(d.)

29

 

We can apply Proposition 1.5.1, to get that the last expression is equal to

fL/R{(( (H— F(Zoqe,+x1,), Emei—l—xL) (1.5)

=1

+tr(B — Pm)DF(Z me,- + xL)}2u(de)d71

1:1
where we assume that l 2 m, K = span{el,...,el},L = ,_1 her 6, (the closed
complement of K in X from Proposition 1.5.1), x1, = PLx, PL IS the projection of

X onto L and 7) denotes the standard Gauss measure on R’.

Using Lemma 1.5.1 we can bound the last expression by

[XIIIIPI — P) )Ia: )IIH + “(P)- PHI)DGFI )Hirmldel

Both components converge to zero as l, m —> 00. Indeed,

AIIIH—PHWIIMIHHIII) = / II ZI ))He.IIH)IIcI:c)

i=m+1

/ Z (e..F a:))HIIIcIa:)

i=m+1

|/\

since F E L2(X,H,du).
Similar argument shows that the second component converges to zero. Resum-

ing, we proved that,

PnF—->F E L2(X,H,du) and
PnDGF—> DGF e L2(X,H®2,du).

We also obtained the following estimate :
/ ILPHFI (mm H.) </ {IIPKF( )IIH + IIPHDGFII: )IIHH)HId:c) (1.6)
for PK E P(’P*).

30

 

Furthermore, the L2(X) limit, LF, does not depend on the choice of the se-
quence of projections. Indeed, let {Pn};,'°:1, {62”}le be two sequences in ’P(X*)

converging to I d H. Then we have,

“LPHF _ LQmFllizw)
< “(10,, _ om)F(H)IIi,(X,H, + H(PH — QH)DGFII)Ili.Ix,H®2)

S 2{||P..F— FHL2(X,H) + lanDGF— DGF||i2(x,H®2)}
+ 2{llQmF “ FHL2(X,H) + llQmDGF —‘ DGFlli2(x,H®2)l

with the RHS converging to zero as m, n —> 00.

The inequality,
l. ILFII)|2II(d-'I) s /,,IIIFII:)I%, + llDGF(m)lliH®2)II(dI:)

follows from (1.6).

Now, Theorem 5.2 in [29] can be extended to our case.

Theorem 1.5.4 Let F E H1(X —> H; du) and to be a positive weight function, i.e.
w : X —> R is measurable, w(x) > 0 Vx E X and w(x+ ) : H —> R is continuous

\7’x E X. Assume that DGF(x) E H‘g’2 for u a.e. x and that
/,,IIIFI:c)IIIH + IIDGFIw)IIHH2)wa)uIdx) < oo.

Then F E D(L). Furthermore, there exists a positive, measurable function It :

X —> R, depending only on to, such that

f, ILFII)I2Ich)H(dm) s L(llflflllt + IIDGFIz)IIin)wa)H(dx) < oo.

31

Proof. The proof in [29] applies if instead of references to Corollary to Theorem
4.2 and to Theorem 5.1 [29], references to Corollary 1.5.1 and to Theorem 1.5.3

are made.

E]

Definition 1.5.7 A measurable map F : Z —> H is said to be in class H — G1 if
the following conditions are satisfied :

(1) V2 E Z 3DF(z) E H692 such that

W2 + h) — F(z) — DF(z)hllH = oIIIhIIH) as IthH —+ o.

(2) Vz E Z, DF(z + ) : H —> H‘g’2 is continuous.

Now we obtain the following Corollary from Theorem 1.5.4.
Corollary 1.5.2 H — C'1 C ’D(L).

Proof. Clearly w(x) = {1 + HF($)“%{ + I]DF(x)|I§,®2}—1 is a weight function
for F E H— Cl. Also H — G1 C H1(X —> H,du). Indeed, first, F E H — G1
implies Fréchet differentiability of F which is stronger than SG-Differentiability.

Also F is strongly measurable in view of separability of H. Further we need strong

measurability of the H -Fréchet derivative of F, DF : X —> L(H). We have,
1
(H. 6 ag e H, g((FCv + th) — F(x)).g) —+ (DF(x))Ih s g).

The LHS of above is measurable, therefore the RHS, as a limit, is measurable.
Thus DF : X —> H ‘32 is weakly measurable. Furthermore, the inclusion H 8’2 ‘—>
L(H) is continuous and H ‘32 is a separable subspace of L(H), giving that DF is
separably valued and weakly measurable as a function with range in L(H). Hence,

it is a strongly measurable map from X to L(H).

32

 

To prove the RAC condition note that, by (2) in Definition 1.5.7
2 “F(x + tI+1h)- F(ft + tIh)||H S sup ”DF(I: + ah)||H®2 llhl|H(b — 0)
i=1 (:6 a,

where a = t1 3 t2 3 g tn+1 = b is a partition ofan interval [a, b]. Now, F(x+th)

is an absolutely continuous H valued function, so that it is RAC.
Cl

Since both, the Ité-Ramer and Ogawa integrals involve a series expansion of
the integrand with respect to one dimensional Wiener integrals, one can expect to
have a connection between these two types of integration. We give our result in

the next Proposition.

Proposition 1.5.3 Let u E H — C'1 and assume that the H -Fréchet derivative
of u, DFu(x), is a trace class operator on H for every x E X. Then u E D(L)
and u is Ogawa integrable with respect to all ONB’s {e,,},i,'°:1 C X * of H and
63(u) = L(u) + trDFu u a.e.

Proof. By Corollary 1.5.2 we already know that u E B(L). Since Lu exists
we can choose any sequence {P1(;}‘,’V°=1 C ’P(X*) of finite dimensional projections
of the form: PN = 2,19; he (8) h], with {hk},°c‘?:1 C X* being an ONB in H and we

have L pNu —> Lu in probability. Compute the expression for L pN (u)

N N
LF,,(H —Z(u( )hk) hk(x —Z(D x)hk,hk) (1.7)

(recall that hk(x) = 11(hk)(x)). The sum defining the Ogawa integral converges in

probability because the two other expressions in (1.7) converge.

33

 

1.5.4 Comparison of Itfi-Ramer and Skorohod Integra-
tion

Let us first consider the problem of relationship between stochastic differentiation
in the sense of Gateaux and Malliavin. It will be used in comparing different types
of stochastic integration. This question was raised by Mandrekar and Zhang in

the concluding remarks of their paper [33].

Proposition 1.5.4 Let F E L2(X) and assume that %(F(x+th) — F(x)), h E H,
converges in L2(X) as t —-> 0. Then the Malliavin derivative D}? F exists and

coincides with the above limit.

Proof. We can apply the method of proof of Proposition 2.2. in [41]. The

following formula is valid for functions fm E H Gm, m = 0, 1, :

I..If..)(x+eh)=i "7 5H.-
i=0 Z

I,-((fm(t1, 25,-, 25,11, tm), h(t,+1)...h(tm)))(x).

This can be justified first for functions fm of the form: fm(t1, ..., tm) = e(tl)...e(tm),
e E H, ||e||H = 1 and then for functions fm(t1, ...,tm) being a symmetrization of
e1(t1)...el(tp,)eg(tp,+1)...ek(tp,+m+pk) with p1 + +1);c = m and e1, ..., ek orthonor-
mal vectors in H. Finally one can use a convergence argument to get the above

formula for all f E H 9m.

From this point we can proceed as in [41] with obvious changes.

[I

As an example for equivalence of SG and Malliavin differentiation consider

elementary processes ([41]) .

34

Definition 1.5.8 A stochastic process u = {ut; t E T} on X is called elementary

if u is of the following form:

Mic) = 2:1 ¢j(61($),~ - .6N(:v))6j(t)

where 61,...,€N E X* and are orthonormal in H, w,- : RN _—-> RN (j = 1,...,N)

are smooth functions with all derivatives of polynomial growth.

Note. An elementary process can be considered as an H -va1ued random variable

onX.

Corollary 1.5.3 Let u be an elementary process. Then the Malliavin and SG and

Fréchet derivatives of u coincide,

Here e(x) = (el(x), ...,eN(x)).

Proof. The reminder in the form of Lagrange in Taylor series expansion of each
w,- is bounded above by a polynomial in ”x“, multiplied by a factor independent
of x and converging to zero as the increment converges to zero. Therefore the
reminder converges to zero in L2(X), hence Proposition 1.5.4 is applicable to each

of the random variables w,- (E) E L2(X). Assertion for process u follows.

Cl

Corollary 1.5.4 Let u E L2(X,H) be an H—valued, SG-Difierentiable random

variable. Assume that the following condition is satisfied:
1
Vic, h E H —((u(x + 5k),h) — (u(x), h)) (G — M)
e

35

converges in L2(X) as e —> 0.
Then the Malliavin derivative Di"! (u, h)(x) exists and equals to (DGu(x)k,h) u
a.e. If SG—derivative DGu(x) is a trace class operator, then,
trDGu(x) = Z Dgflu, en)(x)
n=1

a a.e., for any {en}f;1 an ONB in H.

Proof. Existence of SG-derivative DGu implies that for all k, h E H,

(u(x + ck) — u(x), h)

E

— (DGu(x)(k), h) —> 0

 

in probability a. It follows by Proposition 1.5.4 that under condition {G-M) the

Malliavin derivative Di” (u, h) exists and
(DGUC’BWC), h) = DttIu, MCI?)

outside NW, C X with p(Nkfi) = 0.

Now the last statement of the Corollary follows because H is separable.

E]

Note. We do not claim that the Malliavin derivative of u exists or that it is a
trace class operator.

In view of the Corollary 1.5.4 we propose the following notion:

Definition 1.5.9 An H -valued random variable u E L2(X, H) will be called weakly
(G-M) difierentiable if u is SG diflerentiable and it satisfies condition (G-M).

We obtain our result on relationship between the It6—Ramer and Skorohod in-

tegrals as a conclusion from the above work.

36

Theorem 1.5.5 Let u E L2(X, H) and assume that the Malliavin derivative DM u
exists and u is weakly (G-M) differentiable. Then the Malliavin derivative and
SG-Derivative of u coincide. If in addition, u E H — C’1 then the Malliavin and
H -Fre’chet derivatives of u are the same. Consequently, if for every x E X, DMu(x)
is a Hilbert-Schmidt operator on H, with E || DM u ||§i®2< 00 and u E H —C'1 then
Lu 2 Is(u) a.e. du.

Proof. Since for any k, h E H,
(DMUUCLh) = (DIWUXh) = Dim“) h) = (DGUW, h) II a-e.

we get the equality of derivatives. Equality of integrals follows from (1.3) of Propo-
sition 1.4.1 and (1.7) of Proposition 1.5.3,

[S(PNu) = LPN (u) u a.e.

with I3(PNu) —> Is(u) in L2(X) and LPN (u) —> Lu in probability.

1.5.5 It6—Ramer Integral as an Integration by Part Op-

erator

The idea to exploit an integration by parts formula in the problem of transfor-
mations of Gaussian measures on a Banach space was used by Bell in [2], [3] and
by Ustunel and Zakai in [56]. The question of absolute continuity of the origi-
nal and transformed measures was considered. Therefore it is of an interest to
know whether the It6—Ramer integral operator L satisfies the integration by parts

formula.

37

Let us recall definition of Integration by Parts Operator (IPO) as in Bell [2],

however we use a different class of test functions.

Definition 1.5.10 A linear operator A : D —> L2(X,du), where D C HX, is
called an Integration by Parts Operator ( 1P0) on D for u if the relation

L(DF¢($).U($))II(d-r) = f, ¢(z)(Au)(rc)II(drv)

holds for all 0;on functions 45 : X —> R (continuously Fréchet H—Difierentiable
functions with the Fréchet derivative DF of polynomial growth in directions of H ),

and all u E D for which either side of the above exists.

It will be useful to note that the Ito-Ramer operator L is continuous in the

norm
ll’ull2 = EllUlliq + EIIDGUIlimz- (1-8)
This follows directly from the proof of Theorem 1.5.3.

Theorem 1.5.6 The It6-Ramer operator L is an [PC on the closure in the norm

( 1.8 }, of the linear space of elementary processes in D(L).

Proof. We need to show that for any C501,, function (I) : X —> R and any u E D,

fX¢(w)Lu(w)II(dw) = / IDFIIx).qu))HIdH) (1.9)

x
where D is the closure in the norm (1.8) of the set of elementary processes in D(L).
Let PN = 25:, f). (a f). e P(X*), where {12.33:1 c X* is an ONB in H. We

begin with an elementary process

uiw(x) = Z; w,(e1(x)...eM(x))e,-(t).

Note that in the case of elementary processes,the Fréchet derivative and SG-

derivative coincide. We have the following expression for L [2,, (uM )(x)

 

LP~(UM)($) = g(U-WLfkkllekW) _I;(DGUI($)(fk)Ifk)H
= _ 231M“ €33( ))(ez‘IfIcleII($ )
_ [2:22:12]: (g(x))(eiafk)H(ejafk)I-I

where e(x) = (e1(x), ..., eN(x)) for short.
We want to find the Ito-Ramer integral of uM , that is we need to find an L2(X)
limit of LPN (uM )(x) as N —-> 00. We have the following:

N M M
Eh; ;¢I(é)(€z', fk)ka — Eda-(5)632
M 00
= Eli: ¢i(é) Z (31" fk)ka}2
i=1 II=N+1
M 00
s M2;E{¢I(é) Z IeI.fI)HfI}2.

k=N+1

Each of the M terms in the latter sum converges to zero in L2(X). Also,

N M M a i M a i
_ a: (e(x))IeI. II)HIe.-, II)H — Z 3‘: (5(2))
M M awiw N

8$j(t‘3 $))21(e(iafk)H (eijk)H—(€II€I')H)

 

 

 

converges to zero in L2(X) since it is a finite sum of square integrable random

variables multiplied by non-randomfactors converging to zero. Therefore,

L(HM) : Z II,(H)H, — Z 8331(5).

 

On the other hand,

(DF¢($)I W (93)) = 2(DF¢($)I ¢I(é(iv))eI-).

Thus it is enough to show the following equality:
3 I-
fX¢(m){¢I-(é(x))eI-(r) — a:i(é(:v))}u(dx)

=[x(DFquv),¢I(é($))€z')#(d$)-

 

By Proposition 1.5.1, for K = span{e1, ..., 6M} and K, its closed complement

in X from the Proposition, we have,

— 8‘:IIII))}IIIII~)

i

 

l. IIcc)III.IeIx))eI-Ix)

 

M
Z I H IIIHH + Z eI-IIH + II)eI){II.-IéI:cH + “))H-(II + IX)
X j=1
8 I-
— 51::(e(xk + $K))}#R(d$f<) ® WWW)
M
= fl? [RM ¢<$K + jZ21ajejfllpilozla"'IO‘M)O""
8 I-
_ 8: (H1, HHHIHIdamI-IIIIH)

where a = (a1, ...,aN) E RM, xx :2 29:1 e,v(x)e,-, x1? = x — xK and 7M is the M
dimensional Gauss measure on RM. Using the fact that the divergence operator is

an IPO on RM ([20]) we get, that the last expression equals to,
M
f1? /RM(DF¢($R + 2 (13'8le ¢I(0IiI ---I aMleIl’YM(d&)Mk(d$R)
j=1

= LIDFIIH),I,IH.II), eHIx))e.-)IIIII~)-

Hence we have formula (1.9) for the Ité—Ramer integral L and the class of
elementary processes. Next, if uM ——> u in the norm (1.8), then because the

operator L is continuous in this norm we get LuM —> u in L2(X). This implies,

/¢L’U,Md,UI—)/ quudu and f(DF¢,uM)du—>/(DF¢,u)du.
X X x X

This completes the proof.

40

D

The problem is to show that the closure, in the norm (1.8), of elementary
processes in D(L) coincides or contains the class of processes introduced in Theo-
rem 1.5.3. We do not investigate this question here, however we have one simple

Corollary (due to the IPO property of the Skorohod integral ([33])).

Corollary 1.5.5 L is an [PC on the class of processes u E L2(X,H) satisfy-
ing assumptions of Theorem 1.5.5 when the class of test functions is restricted to

elements Ip for which DMIp = DGIp.

Notice, that the IPO property of the Skorohod integral involves the Malliavin
derivative DM which is the adjoint operator to the integral operator of Skorohod.
Our work indicates that, in the sense of Theorem 1.5.6, the adjoint operator to the

It6-Ramer integral L is the SG-Derivative DG.

1 .6 Examples

1. Elementary processes. Let u be an elementary process (see Definition 1.5.8)
of the form: ut = Zj-Vzl ij(é)e,-(t). Then u is Ito-Ramer integrable and

N N ,
Lu = XIII-(ea.- — Z Z—fje)

j=1 j=1

(see the proof of Theorem 1.5.6). The Ogawa integral of u also exists, 6°(u) =
9:1 lpj (5)63”
2. Brownian motion. Let {Bt,t E [0,1]} be the standard Brownian motion of

Example 1.2.1. Then the process ut 2 f5 Bsds is an H valued stochastic process,

where H is the RKHS of Brownian motion. The Ogawa integral of u is given by

°° °° 1 1 , den
6°(u) = ZIu.eI)I(e.) = ZIB.e:.)I.II,II f0 eidB = ,3? (e. = a)
71:1 11:1

41

as proved in Example 3.2 [52] (the above formula is correct for any ON B {e,,}f,°=1
in H).

Compute the Gateaux derivative of u in the direction of h E H. Since u(x +
h) — u(x) = f0 h(s)ds, we get that DGu(x)h = ID h(s)ds independently of x E X.
Operator DGu(x) on H is Hilbert-Schmidt because it has the same Hilbert-Schmidt
norm as the kernel operator on L2([0, 1]) given by a kernel 1[o,t](S)- Thus it is Ito-

Ramer integrable by Theorem 1.5.3. By Example 3.2 in [52]
N

IIIPNDGIIII» = ZI/ en(s)ds, III-))H

n=1 0
converges as N -—> 00 independently of the choice of an ONB {e,,},j’,°=1 C H. For
the ONB consisting of indefinite integrals of the Haar functions the result is easy
to obtain and it is equal to %' Thus Lu 2 %B12 — %. Ité-Ramer integrability of u
also follows from Corollary 1.5.2 since Du(x + ) : H —> H 8’2 is continuous (it does
not depend on h). Hence, u : X —+ H is an H — G1 map. The Skorohod integral

I 3(u) is the same as the Ito integral f01 BtdBt (see Example 1.3.2). Hence,

3 1 1
I (II) = L(u) = 53? — 5.

Notice that all the above could be also justified in a similar way by use of
Ramer’s and Kusuoka’s results in view of the relationship given in Examples 1.3.2

and 1.3.1.

3. Reversed Brownian motion. Let us now consider the reversed Brownian
motion process B1_t. By Theorem 1.5.3 and arguments as in (2) above, the process
ut 2 f5“ Bl_sds is Ito-Ramer integrable. Also the Skorohod integral I3 (u) exists.
The Malliavin and Gateaux derivatives of process u coincide and are given by
Du(h) = f6 h(1 — s)ds. Theorem 1.5.5 implies that Lu 2 Is(u).

Example in [52] shows that u is not universally Ogawa integrable in the sense

of [52]. Note that convergence in [52] is in L2(Q).

42

4. Ogawa non-integrable process. The following example in [44] shows that
given any ONB in H one can construct a process that is not Ogawa integrable

with respect to this basis. Let

u. = i: n—Z-eIIt)sIgnIvrIe.)) I; < p s 1).

then u E L2“), H), but the series defining 6°(u),
i —sIgnIvrIeH))IrIe.) = i iIIIe.)I
n? n?

n=1 n=1

diverges a.e.

5. An example of a process with an infinite Wiener Chaos expansion.
Consider the general case of a Gaussian process X = {Xh t E T} defined on a

probability space (9, f, P) and the associated triple (i, H, X). Let

M8

In(el(t1)...e1(tn))en(t)

“It 2

3
ll
H

7III(II(€1))6II(I)

II
M8

file file
ara-

3
ll
H

where {en};',°=1 C H is an ONB of H, ’Hn’s are Hermite polynomials normalized as
in Section 1.3.1.
(a) u E L2(Q, H) iffp >2 - ,since

00 l 1 0° 1
EHUHH=E§fi57HIZIW 7H3(1 1)):231335'

(b) u is Ogawa integrable with respect to ONB {e,,}f,"=1 if p > %

We have
glIu,e.)IIe.)=:_:n1, «1va (6 ))«Ie.).

We need to check when this series converges in probability. Since (excluding

the first term) the series consists of centered, integrable random variables adapted

43

to the filtration .71}, = o{7r(e1),7r(e2), ...,7r(e,,)}, (n 2 1), f0 = {0, (2}, it converges

P-a.e. on the following set (see Proposition IV.6.2 in [38]):

Q0 2 {:Eflnp an—l IHM (61))” (en))2lfn—1}<OO}

°°11

= {:1 fi—Hz 61)) < 00}.
But
El: i‘1!"l'LI2I(7r(61))}= Z 512; < 00,

therefore P(Qo) = 1, and

6::Iu)= i;—j——§II H(rIe ))IIe.).

(c) u E D(IS) iff p > %. We need to show L262) convergence of the series defining
the Skorohod integral of u. This can be proved as follows:

236+ 1)III3—field)...eIIIH)eHII))II§,H..I

1:"? p 1”n:1(I,(II)...H1(I.,)H.,,(I)

 

n

+ Z 61(t1)...61(t,_1)el(t)el(t,-+1)...e1(tn)e,,(t,))“2,8,“,qu

— _2 + 272—2— —1;.||e1 616IIIIH®IH+I) — 2 +

n=2 n=2 E; I
Second equality follows from orthogonality of components under norm.

Also, by property (3) of Multiple Wiener Integrals, we get

Iu)=5: fifiw (6))Ir(eI) — 1 = 62(10-

(d) Malliavin derivative.
(d.1) at E D(DM) with DMut E L2(X,H) for t fixed, ifp Z 5

44

For the above to be true the following series must converge.

°° 1 1 °° 1
1 __ 2 = 2
Ennllnp ,__n!{el...el}en(t)||H@n "i=1 n2p—16n(t) < 00,

because 22le e,,(t)2 < oo (en(t) are the Fourier coefficients of C(, t) in H).
(d.2) u E D(DM) with Du E L2(Q, Hm) iffp > 1.

Indeed
oo 1 1 °° 1
!—— 2 n = —< .
£77.77. “n? Mel 618nI|H®( +1) 7121 n2P—1 00

Thus for % < p S 1, u E D(Is) and can be Ogawa integrable with respect to
some basis while Du ¢ L2(X , H ®2).
(e) u E D(DG) ifp > %.
Now we have to restrict our considerations from general probability space (Q, .77, P)
to the triple (2, H, X) and Kolmogorov functional representation of the process X.
This is because the Ito—Ramer integral requires linear and topological structures
on a probability space. Moreover, let us assume that {en}f,°=1 C X" C H.

Fix a: E X. Let 7" E (—1, 1). Denote

1 1

Em

We will use the following estimate for Hermite polynomials (see 7.125 [53]):

UH?) =

’Hn(el(:z: + rh))en(t).

H2n(t) = (—1)"(2n—1)!!et2[c03(/2n+ét+0(1/{‘/fi)]
H2n+1(t) = (—1)”(2n—1)!! 2n+let2[sin(/2n+gt+0(l/\4/R)]

where 0(t) denotes a quantity such that 0—9 = 0(1) and 0(1) denotes a quantity
which is bounded as t —> 00. Note that expressions:

((217. — 1)!!)2/(2n)! and ((Zn —1)!!)2(2n +1)/(2n + 1)! are identical and are of the

45

__1.
2

order n as n —> 00. Hence,

I i: gfiunIelIx) + relIh))enIt)ni. s 0(1) i: :1—

Thus u"(-) : (—1,1)—) H and Zf=1u”(r)—> u(x + Th) in H for every 7‘ E (—1, 1)
and p > i.

By properties of Hermite polynomials, a.e. [u],

 

DG(%—\/l;l_—!7{n(e1)en)(x+rh)(h) = dug)”
1 1
.: fi—nTn’Hwfiefi )+rel(h))(el,h)en

so that du”(r)/dr : (—1, 1) ——) H is continuous. Furthermore,

if iinzflf. Ie (as )+ relIh))Ie1, h)2 s 0(1) SE —,,—_1—,—+—.

2 I
n=171p72 n=171

and the last series converges for p >3 — .Hence the series 20:1 dun (7") /dr converges
uniformly for r E (—1,1). Therefore, with the same proof as for a series of real

valued functions, a.e. [a],

 

: f: n if: $%NHn—1(€1($))(€1ahlen
for p > 2. Note that for p > %, a.e. [,u],
(DC I )h)( ) DMut(x)h = i -n1—P%an—1(ex($))(euh)en(t)

(f) By Theorem 1.5.3 11 E D(L) for p > 1.
Indeed, it E L2(X, H) (proved in (a)) and Dan E L2(X , H ‘32) follows from

E{||DGUII%®2} = wig—5712113. (61)}

 

11121971.
0011
= E < 00.
2p—1
n=171

46

Now assume that p > 4; Then Dan E H‘g’2 a.e. [,u]. Indeed, ||DGu|[H®2 =

°° ——n27-l,2,_1(e1(a:)) converges as in (d.2). Also condition (G-M) holds, be—

n=1 ",2? 71!

cause, as noticed in (e), we even have equality of derivatives. Finally, D. (u, h) H E

L2 (X , H), in view of the following equalities:

EIID.(u,h)II% = En:$%nun 1(61( ))Ie...h)Helu'2

 

Also 21. E D(IS). As it can be seen from (1.3) in the proof of Theorem 1.4.1
and (1.7) in the proof of Proposition 1.5.3, u E D(L). The above reasoning is more
refined than that in [59].

47

Chapter 2

Kinematics of Hilbert Space

Valued Stochastic Motion

2.1 Introduction

In this Chapter we study two topics. First, we want to adopt Nelson’s intuitive
ideas on kinematics of stochastic motion to Hilbert space valued stochastic motion.
The results we obtain show strong relation between Nelson’s regularity assump-
tions for a diffusion and some properties of Doléans measure of some martingale
associated with the diffusion. Next we can see that the Brownian motion process
that arises in this analysis plays similar role as its finite dimensional counterpart
in the analysis of finite dimensional stochastic motion. For this, it is necessary
to modify the stochastic integral of Metivier and Pellaumail [35] with respect to
2—cylindrical martingales. The properties of Doléans measure are used here exten-

sively, which again emphasizes the role of Nelson’s conditions.

48

2.2 Hilbert-Schmidt and Trace Class Operators
on Hilbert Space

In the previous Chapters we identified tensor product of Hilbert spaces, H 8’2, with
Hilbert-Schmidt operators T on H by (Th, 9);; = (T, h ® g)H®2. Now we want to
discuss an analogous identification connected with trace class (or nuclear) operators
on H.

Let us think of H as a unitary space and take H (29 H, a unitary space, with the
usual scalar product defined by (h ® 9, k 8) l)H®H = (h, k)H(g,l)H. Now H”, as a
tensor product of Hilbert spaces, identified with Hilbert-Schmidt operators on H,
is the completion of H (X) H in this usual scalar product.

For any continuous, linear operator T on H with an N dimensional range (N =

1, 2, ...), there exist orthonormal bases {en};°,°=1,{fn}:°=1 C H such that Vh E H,

N"
Th = E ,\,,(h, en)Hf,,, A, > 0,72. = 1, ...N.

n=1
Let us identify such an operator T with the element 2,721 An(en ® fn) E H I8) H
and let us define a norm in H 8) H by

N N
ll 2 Men ® fn)“1 = ||T||1 = Z An
n=1 n=1

where HTH1 denotes the trace class norm of the operator T. The Banach space
H (8)1 H is the completion of the unitary space H <8) H in the norm || - [[1. Since the
completion in the trace class norm of the space of continuous, linear operators on
H with finite dimensional ranges is precisely the space of trace class operators on
H, any element of H ®1 H can be uniquely identified with a trace class operator.

Note that for 9 I8) h E H (8)1 H we have

llg®h||1 = llg®hlln®2 = llgllnllhllH (2-1)

49

and in general || - ”L(H) g [I - “Ham 3 || - [[1. For more details we refer to [19] and
[35].
Identification of the spaces H (8)1 H and H 8’2 with subspaces of the space of linear

operators on a Hilbert space H allows to define symmetric and positive elements

of H (8)1 H and H®2 (see [35]).

Definition 2.2.1 We say that an element b E H (8)1 H, or b E Hm, is sym-
metric (positive) if the associated linear operator is self-adjoint (positive), that

is if (bh,g)H = (h, bg)H, i.e. b = b* where b‘“ denotes the adjoint operator,

2.3 Kinematics of Stochastic Motion

Let H be a separable, real Hilbert space. We assume that {Xt}t61, I 2 [0, T], T >
O, is an H -valued stochastic process defined on some probability space (52, .7, P)
and adapted to an increasing family of o-fields {Ftheh where .7, C .73, Vt E I . For
simplicity we always assume that X0 = 0. Let us recall that a stochastic process
{Xt}t€1 is an H -valued martingale with respect to an increasing family of o—fields
{ft}tEI if Vt E I, X, E L1(Q,}"t,H) and Vs S t, E(Xt|f's) = X, P—a.e.

We introduce, as in Nelson ([36]), the following regularity assumptions on the
stochastic motion X, and, mostly using Nelson’s techniques, we study their conse-

quences.

(R0) t )—> Xt is continuous from I to L1(Q, H)
(R1) Condition (R0) holds and

_ . Xt+A — Xt
BX. — g%{E(—|ft)}

A
exists in L1((Q, E), H) and t +—> DXt is continuous from I to L162, H).

50

Note. DXt defined in condition (R1) can be interpreted as the mean forward
velocity.

With an (R1) process X we will associate the following process:
t
Y, = X, — / stds, te I. (2.2)
0

We will introduce one more regularity condition for a process Y, which may or
may not be associated with an (R1) process X.
(Rm)

__ 8’2

AV, A IE}

 

exists in L1(Q, 7'}, H (8)1 H) and t +—> If”, t 1—> 02(t) are continuous mappings from
I to L1(SZ,H®1 H).
(R23) Same as condition (R24) but with H Q92 replacing H (8)1 H.

We will now show that the mean forward velocity has a similar property as
its analogues: the velocity in a physical phenomenon of motion and the mean
forward velocity in stochastic motion in a finite dimensional space. The latter was

investigated by Nelson in [36].
Theorem 2.3.1 Let {X,hel be an (R1) process. Then for any a g i) with 11,1) E I
E{x, — Xulfu} = E{/ stds|f,}.

Proof. Note that by assumption t I——> X t and t +—+ DXt are continuous mappings
from I to L1(SZ, H) and so is t +—> f3 DXsds.
Let e > 0 be arbitrary. We will prove that

J = {t E [um]: W s s s t ||E{X, — Xu|fu}
_E{/8 Derrlfu}||Ll 3 5(3 _ ,0}

51

is a closed subinterval of [a, v] C I.

Clearly J 75 (I) (u E J) and J is an interval. Denote Tm = sup{t E J} and
we need to show that Tm E J. Let 51 > 0 be arbitrary. Then 35 > 0 such that
VT", — 6 S s g Tm,

51 Tm 51
IIXTm—Xsnm-g and n/ DxrdrnL.<-,-,

by continuity of t )——> X, and t +——> jot Derr.
Hence, for s E J,

Tm
||E{XT... — Xulfu} — E1 / DerrlquIL.
S ||E{XTm — XIII-7:11} "" E‘IXS — XuifuiliLl
+ ||E{X, — Xum} — E{/ DX.dr|f-,,}||L,

Tm
+ HEI/ DerrlquIL.

51

S 2 +e(s—u)+€2—1§el+e(Tm—u).

Since the above holds for arbitrary 51 > O, we get Tm E J.
Now it is enough to show that Tm can not be smaller than '0. If Tm < I) then

377 > 0, Tm + 77 S ’0, such that

||E{XTm+A — XTmlfu} — EfADXTmIquILI

5
S ”E{XTm+A — XTmlme} — E{DXT,,,}||L1 < 2A

and

Tm+A 5
||ADXTm — / Derr||L1 < 5A
T

m

for 0 < A g 77.
First estimate follows from definition of mean forward derivative and contractivity

of conditional expectation (Theorem 4, Chapter V, [15]). Second estimate is a

52

consequence of the fundamental theorem for Bochner integral (Theorem 9, Chapter
II, [15]), in view of continuity of the mapping t )—) DX,.

Therefore
Tm+A
“axe... — mm — EI/ wanna

Tm+A
s IIEIXTm+A — XTmln} — E{/T Derrlf-quL.
+e(Tm — u) S e[(Tm + A) — n].

Thus Tm < Tm + 77 E J which can not happen.

Next theorem is an immediate consequence of Theorem 2.3.1.

Theorem 2.3.2 Let {X,},61 has property (R1), then Y, = X, — f5 DXsds is an

H -valned martingale.
We also have the following connection between the processes Y and 02.

Theorem 2.3.3 Let Y be as in Theorem 2.3.2 and has either property (RN) or
(R22). Let u S '0, 11,2) E I, then

E{(Yv — Yuma} = EI/v rams}.
Proof. Because of the martingale property of Y,
E{(Yv - Yu)®2|fu} = 53fo2 — Yu®2|ful

and we are exactly in the same situation as in Theorem 2.3.1, namely t +—) Y,®2,
t H DY,®2 = 02(t) and t I—-> f3 02(s)ds are all continuous mappings from I to
either L1(SZ, H ®1 H) or L1(Q, H 592). Hence the assertion follows from the proof
of Theorem 2.3.1.

53

[:1

Corollary 2.3.1 Let Y and 02 be as in Theorem 2.3.3 but we restrict ourselves

to the case of (R21) processes only. Then {||Y,||§,},€T is an {R1 ) process with
DllYtlliz = “772(10-

Hence
E{||Yv||iq — manila} = E{/u woman},

forug’v, u,vEI.

Proof. With an element h 8) h E H <8) H we associated a nuclear operator b
and the following holds:

|tr(h ® h)|=lt7‘b|S||b||1=llh ‘8 hl|1= llhlliz,

the inequality being valid for any trace class operator. Therefore

(Y,+A — Y2)” 2
A IE} — 0 (WM

(Y,+A — Ytlm 2
A IE} '— 0 (75))”

_ Y 2
A "'Hla} —tr02(t)|}

Y 2 — Y 2
= MEI“ ”M'HA “ "'Hla} —thIt)nL.Im

 

E{IIE{

 

Z E{ltr(E{

= E{|E{ “Y“

 

 

The last equality follows from martingale property of Y. Since Y is an (RN)
process the last expression converges to zero as A \, 0. Also, because |trT| S ”T“,
for a trace class operator T, the mapping t I——> DHY,||%, = tro2 (t) is continuous from

I to L1(Q). Finally, the mapping t +—> [|Y,||%, from I to L1(Q) is continuous because

EU HYtHIi—“YSHIJH = E{| ||K®2||1—||Ys®2||1|}
< E{||Yt®2-Ys®2||1}

and the mapping t l—> Yf’2 is assumed to be continuous from I to L1(§2, H (8)1 H).

The last assertion in the Corollary follows from Theorem 2.3.1.
B

One can show that Nelson’s requirement on existence of the process 02 can be
expressed directly in terms of process X, namely one can assume convergence of
E{%(X,+A — X,)®2|f,} either in L1(Q,H (8)1 H) or L1(Q,H®2). Because in this
case, both X‘g’2 and Y‘82 are integrable and ”h (8) hIIH®1H = IIhIIH®2 = “h“?! (by
equality (2.1)), adding an extra assumption on DX, to be in L2 (9, H) is reasonable.

Let us formulate our assertion.

Theorem 2.3.4 Assume that {X,},61 and {Y,},€1 satisfy assumption (R21), where
Y, = X, — f0t DXsds. We also require that the mapping t I-—> DX, be continuous
from I to L2(S2, H). The the following statements are equivalent:

(Xt+A — th®2
A

(1) El
(2) E{ (YHA; Yt)®2 [77,} —> 02(t) in L162, H (8)1 H) as A ‘3, O.

 

IE} —> W) in L162. H o1 H) as A \. 0.

 

If the limits exist then B(t) = 02(t) P-a.e. The above remains correct if the as-

sumption (Ru) and the space L1(Q, H (8)1 H) are replaced with assumption (R23)
and space L1(Q, Hm).

Proof. Consider

(Yt+A — Ytlm
t+A
= (Xt+A — th®2 — (Xt+A — Xt) ‘8 t DX,

t+A t+A
— wads e) (x,+A — X,) + ( DX,)®2.

t t

55

Regardless of (l) and (2),

1 t+A W 1 t+A @2
X“ stds) =A(Z/t stds) +0

in L1(Q, H (8)1 H) as A —+ 0. Indeed,
I t+A m 1 t+A 2
E{||A(-A—(/t DX.ds) 11mm} = AE{||Z i stdsnfl} —+ 0

since the mapping t —> DX, is continuous from I to L2(Q, H). Now consider the
second term in the expansion of (Y,+A -— Y,)®2. Under the assumption (Rm) on X

we have by equality (2.1)

1 t+A
IEIHIXHA — XI) <2) (X /, DX.ds)HH®.H}>2

1 t+A
s EIIIX... — xtnimnz /, stdsllé}

1 t+A
= EIHIXHA — XI)®ZIIH®.H}EIHZ ft DXsdslliJ}

and, as A \, O, the first factor converges to zero in view of the assumption (R2,)
while the second factor converges to E{||DX,||}°1,} because the mapping t )—> DX,
is continuous from I to L2(Q, H). The same argument works for the third term
in the expansion of (Y,+A — Y,)®2. Hence the equivalence of ( 1) and (2) follows by
contractivity of conditional expectation. The last assertion of the Theorem follows

from equality (2.1).

2.4 Stochastic Integration in Hilbert Space

Stochastic integration in Hilbert and Banach spaces is a subject of the monograph

[35]. We recall here the isometric integral and we explain why it is not a sufficient

56

tool for solving the problem of recovering the noise from stochastic motion in
Hilbert space. Then we recall the concept of 2~cylindrical martingales and, using
the main ideas in [35], Chapter 16, we introduce a stochastic integral with respect
to an H-valued (R21) martingale. This stochastic integral admits a wider class of
processes as integrands than the isometric integral and the cylindrical integral of
[35], Chapter 16. We use the results of this section to give a partial answer to the

question of the role of Brownian motion in stochastic motion in Hilbert space.

2.4.1 General Assumptions and their Consequences

In what follows we always assume that the filtration {Jihe 1=[0,T] satisfies usual
conditions, i.e. that:
(1) The filtration is right-continuous, i.e. Vt E I, f, = 03),.733.
(2) The probability space (9,75%, P) is complete and Vt E I, .7, contains all sets
of P—measure zero, which belong to .7}.

Two processes X and Y are said to be P-equivalent if P({w : Elt,X,(w) 74
m2») = o.

A stochastic process X is called cadlag (in French: continue ‘a (_iroite et admet
une limite a. gauche) if Va) E Q the sample path t +—-> X, (w) is right continuous and

has left limits.

Definition 2.4.1 . We define M%, the space of H ~valued, cadlag, square inte-
grable martingales (i.e. E{||MT]|%{} < 00, which implies that
sup,€1E{||M,||§,} < co) and we identify P-equivalent processes. In the case of

H = R we will write M%(R) to avoid a possible confusion.

As it is shown in [35], Section 10.1, M? is a Hilbert space with a scalar prod-
uct defined by (M, N) M? = E{(MT,NT)H}, because there is a one-to-one cor-

57

respondence (up to P—equivalence) between H -valued, cadlag, square integrable

martingales and elements of L2 (S2, .Fp, H).

Remark 2.4.1 If Y, = X, — Id DXsds with X an (R1) process then there exists
a version Y’ of the process Y {i.e. Vt E I, P(Y, = Y,’) = 1) which is cadlag.
Moreover, if E{||YT||§,} < 00 then Y’ E M? and Y and Y’ are P-equivalent.

Indeed, Y is an L1(Q, H) continuous martingale and therefore it has a cadlag

version by Theorem D-6 in [50].

In view of the last Remark, from now on we assume that Y, = X, — f5 DXsds
is a cadlag martingale and if Y is an (Rm) or (R22) process then Y E M? follows

from equality (2.1).

2.4.2 Doléans Measure of (R2,) Elements of M],

First we recall basic definitions and properties of Doléans measure in general.
Let us recall our general assumption that {X,},EI=[0,T] is an H -valued stochastic
process, where H is a separable Hilbert space.

A set A = F x (s, t] C {2 x I, where F E f, is called a predictable rectangle
and the collection of predictable rectangles is denoted here by 3?. The o-field
generated by ER is called the o-field of predictable sets and denoted by ’P. A
stochastic process is called predictable if it is 73 measurable.

Assume that for a process {X,},EI, X, E L162, H), Vt E I. For each A =
F x (s,t] E 3? define

«m=mMmer.

If a extends to a o-additive, H -valued measure on ’P, then it is called Doléans

58

measure of process X. The following results on Doléans measure are proved in

[35], Sections 2.6 and 14.3.

Theorem 2.4.1 Let M E M%, i.e. it is a cadlag, square integrable martingale
with values in a separable Hilbert space H. Then:

(I) {“Mtllirhel has Doléans measure, which will be denoted by 04an.

(2) {M92},61 has Doléans measure with values in the set of positive, symmetric
elements of H (X), H (see Definition 2.2.1). We will denote this measure by aM.
Moreover,

O‘IIMII = th = lam!

where | - | denotes variation of a measure.
{3) There exists a unique, up to allMll equivalence, predictable H (8)1 H -valued process

QM, such that
aM(G) = [G QMdaHM”, voe P.
The process QM takes its values in the set of positive, symmetric elements of
H (X), H and

ter(w,t) = llQm(w,t)llH®.H = 1 0-6- O‘IIMII-

Now we will see how Nelson’s regularity assumptions interfere with properties of
Doléans measure. Because Doléans measure of an M? martingale takes its values
in trace class operators on H, from now on we restrict our considerations to (Rm)

processes only.

Theorem 2.4.2 Let X be an (R1) process and Y, = X, — f5 DXsds be an (R2,)
element of M2}. Then, with the notation of previous sections:

(1 ) There exists a jointly f (8) 3(1) measurable versions of D||Y||§1 and 02.

59

Let us further consider these jointly measurable versions and let us denote these
versions by the same symbols. Also let us denote by EP®A expectation with respect

to the measure P <8) A, where A is Lebesgue measure on I.
(2) Doléans measure any” of the process “Y”2 has density
Ep®,\{D[|Y|]%,|P} = Ep®,\(tr02[P} with respect to the measure P <8) A,

dam = Ep®,{tro2|P}d(P s A).

( 3) Doléans measure ozy of the process Y‘g’2 has density Ep®,\{02|P} with respect

to the measure P (X) A,
Clay 2 EP®A{02]p}d(P ® A).

and its density, Qy, with respect to the measure any” satisfies the following equa-
tion:

02 = Qytro2 a.e. P (8) A.

Proof. (1) Note that the mapping t )—> 02(t) is continuous from I to L1 (S2, H (8),
H) and hence, the mapping t H tr02(t) is continuous from I to L1(§2). Therefore
(1) follows by Theorem 1.2 in [16].

(2) For predictable rectangles F x (s, t] E 3? we have

O‘IIYII(F X (8, 15]) = E{1F(||Yt||i; — Ill/HIE}
= E{1p f: tr02(r)dr}
= foM ”0201(1)? (g) A)
The expressions at the beginning and at the end of the equality, both extend to

measures on P and these extensions agree on generators, 3?, of P, hence they are

identical.

60

(3) An analogous equality as in the proof of (2) holds also here and the same

extension argument can be applied in this case to yield
day = Ep®,\{02|P}d(P <8) A).

Now, we have the following situation: ay << any” < P (8) A and therefore

day = day dallYll
W” 69 A) danvu 40’ <8 A)

Indeed, let {h,,}f,°=1 be a dense subset of H. Denote

 

day K. 2 day daily”
d(P ® A), dally" d(P ® A)

 

0 =
and define

A, = U{((9 — n)h,,,hm),, > 0}, A_ = U{((6 — n)h,,,hm)H < 0}.

Because

fA ((9 — thn, hmlHdUD 69 A) =/ ((9 — K.)h,,, hm)Hd(P 8) A) = 0

we get that PIX) A a.e. Vn,m = 1, 2..., ((6’ — s)h,,, hm)” = O, implying I9 _-: It P®A

a.e.

2.4.3 Inadequacy of the Isometric Stochastic Integral

We recall the isometric stochastic integral of Metivier and Pellaumail [35] with
respect to martingales from Mg. We begin with defining the class of integrands,
however we will restrict ourselves to processes with values only in linear operators
on H. For a slightly more general case of operators from one Hilbert space H to

another Hilbert space K we refer to [35].

61

According to Theorem 2.4.1, with a martingale M E Mg, we can uniquely as-
sociate predictable, H ®1 H -valued process QM. We can think about Q M as taking
its values in the space of trace class operators on H using the usual identification
of Section 2.2. Since the values of QM are actually self-adjoint, positive operators,
there exists a square-root, denoted by Q32, which is a Hilbert-Schmidt operator

and is defined by Q12.” 0 Q15” 2 QM.

Definition 2.4.2 Let M E M? We call L*(H,P,M) the space of processes X,
the values of which are ( possibly non-continuous) linear operators on H, with the
following properties:

{1) For every (w,t) E Q X I, the domain D(X(w,t)) of X(w,t) contains Qf],(H).
{2) For every h E H the H —valued process X o Qifih) is predictable.

(3) For every (w, t) E Q X I, X(w, t) o Qifiw, t) is a Hilbert-Schmidt operator and

_1_
[W “X o Qiuim < oo.

Proposition 2.4.1 For every X, Y E L*(H,P, M) the process X 062M oY* takes

its values in trace class operators on H, it is predictable and

/ tr(X o QM o Y*)da“M” < 00.
9x1

The bilinear form (X, Y) +—> fgxltr(X o QM o Y*)doz"M” is a scalar product on
L*(H,P, M) and for this scalar product this space complete.

A process X is called elementary if it is of the following form:
X(w,t) = fail/41'0”) t) (2.3)
i=1
where u,,i = 1, ..., n are continuous, linear operators on H and {A,-},?=1 C 5R.
Note. We will always assume that if A, = F,- x (3,, t,], A,- = F, x (3,, t,] and i ¢ 3',

62

then (3,, ti] 0 (Sj, tj] = (b by taking more refined partition of I if necessary.Observe

that elementary processes are in L*(H, ’P, M).

Notation. The closure of the space of elementary processes in L*(H, ’P, M) will
be denoted by A2(H,’P, M). Let M E M? The isometric stochastic integral
is the unique isometric linear mapping from A2(H, ’P, M) into Mg, such that the
image of X = 1Fx(s,,]u, for every predictable rectangle F x (s,r] and continuous

linear operator u E L(H), is the martingale {1p[u(MMt) — u(MsAt)]}t€;.

We conclude this section with an example motivating extension of the isomet-
ric stochastic integral. Nelson’s idea to recover the noise from stochastic motion

described by a process X was to compute
t
Wt 2/ o_1(s)dYs.
0

Under some conditions the process W turned out to be a Brownian motion. Also

the stochastic motion X would satisfy the following stochastic integral equation

t t
X, =X0+ / DXSds+ / o(s)dWS
0 O

1 existed and were an admissible

(see Paragraph 11 in [36]). In our case, if 0"
process for the isometric stochastic integral then we would get Wt = f0t o‘1(s)dY, E
M? However this does not happen and we give an example explaining why the
isometric stochastic integral is not a sufficient tool for Nelson’s technique.

Let us make some regularity assumptions about the process 02.

Definition 2.4.3 We will call the process 02 regular if.
(1 ) 02 is a predictable process which takes its values in positive, self-adjoint ele-

ments ofH ®1 H.

63

(2) V(w, t) E QXI all eigenvalues An(w, t) of 02(w, t) are strictly positive, An(w, t) >
0, n = 1, 2...

Thus for a regular process 02, for every (w, t) E Q x I, there exists an orthonor-
mal basis {hn}f,°=1 C H such that
02(w,t)(h) = Z A..(w.t)(h, h..(w.t))Hh..(w.t). Vh e H
n=1
with An(w,t) > O, n = 1,2..., 230:1 An(w,t) : ||02(w,t)||1.
Also, there exists the square-root of 02, denoted by a, which is a Hilbert-Schmidt

operator (see [35]) and has the following representation:

0(h) = E @(h, h,),,h,,, Vh e H
n=1

(we will usually drop the dependence on (w, t)).

2

Note. From now on we will always assume that the process a is regular.

The generalized inverse of o ([32]), denoted by 0‘, is defined by a composition
P[K6,(a)]1 o 0‘1 o Pcl(Ran(0))v where P[Ke,.(o.)].1. and PCNRGMU» are respectively projec-
tions on the orthogonal complement of the kernel space and on the closure of the
range of o and 0—1 is the inverse relation to the operator 0. Note that because 02

is regular, cl(Ran(o)) = H. Then 0‘ takes the following form:
OO 1

_ h = —— h hn hn Vh R .

0’ ( ) ;m( a )H 6 (171(0)

It follows from Theorem 2.10, Corollary 2.13 in [32] and regularity of 02 that o
and o‘ are predictable processes. By Theorem 2.4.2 we have

0.2

 

Qy

tro2

64

 

Hence, D(o'(w, t)) D Ran(o) = Qé(H).
Moreover, a" 0 Q; is a predictable process, so that requirements (1) and (2) of

Definition 2.4.2 are satisfied. However (see (2.12) in [32]),
1 1 1

1
0—0 §=———o’oo=——P era =—
QY x/tro2 \/tro2 [K ( )li x/tro2

is not a Hilbert-Schmidt operator unless H is finite dimensional. Thus 0‘ ¢

A2(H,’P, Y).

IdH

2.4.4 Cylindrical Stochastic Integration

We learned in Section 2.4.3 that the requirements imposed on integrands by the
isometric integral are too restrictive if one wishes to recover the noise from a
stochastic motion in a Hilbert space by using Nelson’s technique. We want to
preserve an (R21) martingale as an integrator, therefore our primary goal is to
increase the class of integrands, to include the process 0‘. Failure of the isometric
stochastic integral in Nelson’s procedure is due to non-existence of a standard
H -valued Brownian motion. In order to realize a Brownian motion process with
covariance associated with an identity operator on H one has to abandon H -valued
processes. It turns out that one can solve this problem with help of cylindrical
processes. It is enough for our purposes to study 2—cylindrical H -martingales,
with H - a separable Hilbert space. For the full theory we refer to [35]. Even
though we are mainly interested in stochastic integration with respect to an (Rm)
martingale, now treated as a 2—cylindrica1 H -martinga.le, eventually we want to be
able to integrate with respect to cylindrical Brownian motion. Therefore we recall

the definition of stochastic integral in full generality.

Definition 2.4.4 (1) A 2-cylindrical L2(SZ,.7:)-valued H ~random element (I is a

continuous linear mapping from H to L2(Q, .7).

65

(2) We call {Mt}tEI a 2- cylindrical H—martingale if each Mt is a 2—cylindrical,
L2 (9, ft)-valued H—random element and Vh E H the real valued process {Mt(h)}t€1

is a martingale relative to {E}t€[.

Note. The space of 2-cylindrical H -martingales can be identified with the space
L(H, M?(R))-
Next we will recall definition of the quadratic Doléans measure of a 2-cylindrical

H -martinga1e.

Definition 2.4.5 For a 2-cylindrical H -martingale M, the quadratic Doléans
function d M is the additive,(H (8)1 H )*-valued function on 3? defined by

(1?: WW X (s, 75D) = E{1F(Mt ® MM) - Ms ® 1913(5)”

where, for every t E I, M, (8) M, denotes the continuous linear mapping from
H (8)1 H into L1(Q,.7:t) which is the linear continuous extension of the mapping
b = h®g I—> Mt(h)Mt(g). Also, above, b E H®1 H, F E ft, s,t E I, s S t.

If d M extends to a o-additive measure on ’P then the extension is called quadratic

Doléans measure of the 2-cylindrical H -martingale M and will be denoted by 05M-

A simple condition for the existence of quadratic Doléans measure for a 2-
cylindrical H-martingale M is that for all h E H, M (h) had a cadlag version ([35]).
Note that it assures existence of Doléans measure for a 2-cylindrical martingale

associated with a martingale M E M? by Mt(h) 2 (Mt, h)H, Vh E H.
Example 2.4.1 Cylindrical Brownian motion.

Let us recall (see Proposition 4.11 in [35]) that the H -valued Brownian motion W
has covariance C E H®1 H, i.e. Vh®g E H(X) H,

(E{Wt®2}, h <8 g)H®2 = “M ® 9)-

66

C, being a trace class operator, cannot be an identity on infinite dimensional
Hilbert space H. If one wants to have a Brownian motion with C = I dH, one has
to consider cylindrical processes.

We say that a process {Wt}t61 is a cylindrical Brownian motion if:

(1) Vh E H, {VI/t(h)}t€1 is a Brownian motion.
(2) Vh,g E H, t E I, E{Wt(h)Wt(g)} = tC(h,g) Where C is a continuous bilinear
form on H x H.

Note, that given any continuous bilinear form C on H x H, there exists cylin-
drical Brownian motion with C as its covariance - see Paragraph 15.4 in [35]. In
the case of C (h, g) = (h, g) H, C is associated with an identity operator I (11; and
we call the cylindrical Brownian motion standard. Note that standard cylindrical

Brownian motion cannot be associated with any ordinary sense H -valued process.

Now we recall definition of cylindrical stochastic integral. We begin with a

proposition which is an analogue of Theorem 2.4.1.

Proposition 2.4.2 Let M be a 2-cylindrical H —martingale and 011;, its quadratic
Doléans measure with bounded variation [a,-4|. There exists a process QM with
values in the set ofpositive elements of (H (8)1 H)* (i.e. QM(h (8) h) 2 O, Vh E H),
such that for every b E H (8)1 H the real process (b, Q M) is measurable for the
lam-completion of the o-field P, it is defined up to [aMl-equivalence and has the

property
<b,a.~.(A)> = l. a.e.-4w» lawman (2.4)

VbEH®1H,AEP.

67

Now, if X is an elementary process of the form (2.3), we define for every

h e H,
(/ xenw) = ileumuxvm — (mummi (2.5)

where u* denotes the adjoint operator. The integral, (f X dM ), is a
2-cylindrical H martingale and for every h E H the real valued square integrable
martingale (f XdM)(h) E M§(R) has norm given by (see 16.2.2 in [35])

”(f XdM)(h)llie(R) = / <X*<h> 69 mm. on diael. (2.6)

Definition 2.4.6 (A) L(M, H) is the set of processes X with the following prop-
erties:

(1) V(w,t) E Q X I, X(w,t) is a linear operator on H with domain D(X(w,t))
dense in H.

(2) Denoting by X*(w, t) the adjoint of X(w, t), the linear form

<X*(w,t)(h) ® X’(w,t)(g)aQM(wat)>

has [aM|-a.e. a unique continuous extension to H X H which results in a predictable
process.

(3)

we = "$55 <X*<h> e X*(h),QM(w,t)> diaelié < oo.

We define A(M,H) - the closure of the class of elementary processes of the
form (2.3) in the space L(M, H).

(B) The unique extension of the isometric mapping X i—> (f X dM ) given by ( 2. 5 ),
from the space of elementary processes into the space of 2—cylindrical H -martingales,
to the isometric mapping from A(M, H) into the space of 2—cylindrical H -martingales
is called the stochastic integral and is denoted again by X H (f X419).

68

Now we want to take advantage of the fact that the integrator in the cylindrical
stochastic integral, which we consider, is actually a square integrable martingale.
We therefore are able to express quadratic Doléans measure, its variation as well as
the process Q M associated with the integrator in terms of its Doléans measure and
the process Q M, which are simpler objects. If we apply this analysis to an (Rm) in-
tegrator, the situation simplifies even more by use of our results from Section 2.4.2.
Note that again, this is a consequence of Nelson’s regularity assumptions on the

stochastic motion.

Lemma 2.4.1 The Doléans measure of a martingale M E M? and quadratic

Doléans measure of M coincide as (H (81 H )*-valued measures on P.

Proof. Indeed, first note that M282 E L1((SZ,.7-}); (H (81 H )*) This is because
if T E H ®1 H then T(h (8 g) = (Th,g)H extends uniquely to an element of
(H (8)1 H)* with ||T||(H®,H)e S ||T||1 (see Paragraph 14.2 (2) in [35]). Therefore
M392 is integrable. Also,

h <59 9 H Mt®2(h ‘3 9) = (Mt, h)H(Mtag)H = M, 8’ Mt“ 8’ 9)-

Hence M592 = M, (8) Mt as elements of L1((Q,.7:t); (H (8)1 H)*). Therefore \7’b E
H®1H, FEft, s,t E I, s S t, we have,

(b, dM(F X (5: till : E{1F(Mt ‘8 Mtlb) _ Ms (8) M803)»

3 2)}

2)})

2 E{1F (b, M592 — M®
= (b, E{1F(M,®2 — M?
= (b, aM(F x (s,t])).

Note that on is an H (8)1 H -valued measure and can be treated as an (H (8)1 H )*-

valued measure. Because M is a 2-cylindrical martingale associated with M E M?,

69

dM on the LHS of the above expression extends to (11,3, therefore QM = on as

(H ®1 H )*-valued measures on P.
C]

Now we explain how a cylindrical integral with respect to a square integrable
martingale can be computed using only the Doléans measure and the associated
process QM. In the case of a square integrable martingale we know that 01M = a M

as (H (8)1 H )"‘-valued measures. Thus

(5, 011C404» = (ham/1)) = <b9AQMda|IMII> = A (b: QM) dauMn,

(we denote an element of H (291 H and its extension to (H (8)1 H )* by the same

symbol) because

<h ’8 9’ [A QMda”M“> Z (A QMdallMll(h)a 9)H
: L(QMWLQMCIGHM”

[401 ® 9, QM) dauMu

so that operation of extension to element of (H (8)1 H )* and integration are inter-

changeable. Also

]aM[(H®1H)* = [01M](H®1H)* S [01M] : O‘IIMII’

where, to avoid a confusion, we denoted by |-|(H®,H). the variation of an (H (81 H )*
valued measure.

On the other hand if ”T“(H®1 H)..- = 0 then, by uniqueness of the extension, also
||T||1 = O. This gives that if [a,-4|(A) = 0 then [aM|(A) = 0f||M||(A) = 0. Thus we

arrived at the conclusion that

laMl(H®1H)* E aHMII-

70

Further,

dlanl

 

(b, (rm/4)) = /A (b, QM) dlaMl = [1(1),in dO‘IIMII

dawn

so that we can choose
_ daIIMn

Q —QM

M ‘ dial
to be a predictable process.
We conclude that if in Definition 2.4.6 we replace the process Q M with QM and

the measure [01M] with mm" to get its condition (2) to hold for

(X*(wat)(h) ® X*(w,t)(g), Qm(w,t)>

and measure CY||M|| it will not change the space L(M, H). Moreover, the seminorm

N<X> = sup{ <X*<h)eX*<h>,oM(w,t>>danMu}%,

llhllsl 9’”

the space of integrable processes A(M , H) together with the stochastic integral, all
remain unchanged. Thus we can integrate processes from the space A(M, H) with

respect to an element M E M? in the sense of cylindrical stochastic integration.

2.4.5 An Example Motivating Modification of the Cylin-

drical Stochastic Integral

In Section 2.4.3 we have seen that o‘ E A2(H,P,Y). The problem of non-
admissibility of 0‘ extends to the cylindrical case. Recall that 02 (see Defini-

tion 2.4.3) is assumed to be regular.

Lemma 2.4.2 For an (R21) process Y E M? we have a" E L(I’, H).

71

Proof. We need to verify conditions (1)-(3) in part (A) of Definition 2.4.6.
Condition ( 1) is satisfied easily since D(o’) D Ran(o).
For condition (2) let us notice that Vh, g E D(o‘) we have,

(9 22—171—_(h )H(gahn)H = (0_(9)ah)H-

Therefore D(o‘) C D((o_)*). Now V(g, h) E D(o') x D(o’) we obtain

 

(we a <<a->*(g>®<a->*<h>.c2y>=(fame-(I0. 02)

tro2
= 1 (02(o_(g)),0-(h))H = L(9t th

tro2 tro2

 

clearly extends continuously to H x H and this extension is predictable in view of
predictability of 02.
For justification of condition (3) let us compute
Nor-)2 = sup {/ <(a->*<h> e («r-m), Qt) dent/u}
Ilhllsl 9x1

1
= sup { [|h llH— —tro 2cl(P (8) A) = /\(I) < oo.
”hug 9x1 WU

As a consequence of regularity of 02 we obtain,

Corollary 2.4.1 Let 0;,(h) = 2713:, 713M", h)Hh,,. Then,
0;, e L2((Q x I,P,auyn); L(H)) c [\(17, H).

Proof. We have

Axlllafillhmdalh’ll s fa (eup{_ })ZAd<P®A>

XI n<N An :1

g/XI( (1)+tro d(P®)\)<oo

72

since Doléans measure of a square integrable martingale has bounded variation.
The assertion follows because L2((Q x I,P,a"y”);L(H)) C 11(17, H) in view of
Proposition in Paragraph 16.3 of [35].

Example 2.4.2 o— 9! AG}, H).

First let us note that

Marv — 0-) = sup (f f: (ht tram? e W = MI)?
IIhIIS1 QXIi=N+1

Thus 01]} does not converge to o’ in the seminorm N.
Assume that 0‘ E A(1~/, H), so that there exists a sequence {Xn }f,°=1 of elemen-
tary processes with N(Xn — o") —> 0 as n —> oo. Denote by PN the orthogonal

projection in H on the span{h1, hg, ..., hN}. We have the following:

N(XnOPN—0'E-)
2

= sup {/M ((X. 0 PN — are) e (X. 0 PN — 0mm. —"—)

“hug “‘02
x tr02d(P (8) A)}

:||h||<1O(nX OPN _ 0N) (hllliflflpéb /\)}

=||h|l<1{ mhz’X; 00)” — (he, h)led(P so A)}

< 811p Whi,X*(l'l/))H — (hi,h)H]2d(P® )0}

llhll<1

PM“;
lg
= sup {/W((1X, — o‘)*(h) a (X. — a’)*(h), Qy) dawn

||h||_<_1

=N(Xn—o_)—>0asn—>oo.

73

We used the fact that for any h, g E D(U'),

((X, — e-)*(h) e (X. — a‘)*(g),Qy)
= —1-—((0' O Xn — IdH)*(h), (0' O Xn — IdH)*(g))H

tro2

and the bilinear form on the LHS extends uniquely to the continuous bilinear form
on the RHS which is well defined on all of H X H. Note that we proved the

following inequality:
N(X,, 0 PN — 0,7,) S N(X,, — o") Vn,N = 1,2...
Next we will prove that for any n = 1,2..., N(X,, 0 PN —> X”) as N —> oo,

Sllp{ ((Xn 0 PN _ Xn)*(h) ® (X11 0 PN _ Xn)*(h)a QY) daIIY”
llhllsl “XI

= sup {/W H 2: mm, X;:<h>)eh. — 5': We, X;(h))Hh.-lltd(P e a}

llhllsl

2 ”3,32%“ g; we, X.:(h>)i.d(P ® M}

sIIXaIEW/Q I Z A.d(P®A)->0
X i=N+1

by monotone convergence theorem. Finally,

N(o,§ — 0—)

|/\

N(o,§ —X,,0PN)+N(XnoPN—Xn)+N(Xn—o')
< 2N(Xn —o—) +N(X,,OPN —X,,).

For any given 5, we can choose an n, such that N(X.,, -— 0—) < e and then 3N0
VN > No, N(Xn 0 PN — Xn) < 5. But this is in contradiction with what we proved
in the beginning of this Example. Hence 0‘ E 1107, H).

74

2.5 Extension of the Cylindrical Stochastic In-
tegral and Application to Nelson’s problem

As we have seen in Section 2.4.3, 0‘ failed to be an admissible process for the
isometric integral. We need to find a larger space of which 0‘, assumed regular,

is an element. Motivation for further studies comes from the following Lemma.

Lemma 2.5.1 For every h E H,

fax] ((01? _ 0—)“) 8’ (”III - 0—)(h), QY) dozIIYlI —> 0 as N —+ 00.

Proof. V(w, t) E Q X I,
<th ((0.; — own 69 (a:v — a-xh). Qy>><w.t> = i (h. mat a o

i=N+1

and is bounded by ||h||§i independently of (w, t).

Now we consider an extension of the cylindrical stochastic integral.

Definition 2.5.1 Let M be a 2-cylindrical H —martingale with the Doléans measure
of finite variation.

(A) Define L'”(M,H) as the set of processes satisfying conditions {1) and (2) of
Part (A) of Definition 2.4.6 and the following condition:

(3) Vh e H, NrIX) = lfnxl <X*(h) e X*(h). an) dlanlfi < oo.
For every h E H, Ni" is a seminorm and we say that a sequence {X,,}f,"=1 C
PM, H) converges to X e from, H) if VII, 6 H N,’;”(X,, — X) —> 0. We will

denote this convergence by Xn => X.

75

(B) We denote by Aw(M,H) the closure in L“’(M, H) of the class of elementary
processes in the topology of convergence ” => ” defined in (A)-(3).

For everyX E Aw(M,H), h E H we define (f XdM)"’(h) as a limit of(andM)(h)
in MflR), where Xn => X and X,, are elementary processes. We call (f X dM )w E
L(H, M§(R)) the stochastic integral. ’

Note. We need to justify correctness of Part (B) of the above Definition. First,

for any sequence {X.,,}f,"__=1 of elementary processes, such that Xn => X we have

VhEH,

|l(/XndM)(h)—(medM)(h)lle;(R) = ||(/(Xn—Xm)dM)(h)l|M3(R)
= N,;”(X,, —Xm)

by equality (2.6). Therefore whenever Xn => X, then Vh E H, {X,,},i’,°=1 is a
Cauchy sequence for NZ” and hence, (f X,,dM)(h) converges in M%(R) to a square
integrable real valued martingale.

Now, the mappings h —> (f XndM)(h) from H to M%(R) are linear, continu—
ous and for every h E H there exists a limit, which we denote by (f XdM)"’(h).
Therefore, by Banach—Steinhaus theorem ( f X dM )w(h) E L(H,M§(R)) - that
means (f X dM )1“ is a 2-cylindrical H -martingale.

We conclude our considerations on Nelson’s ideas with an analogue of Theorem
11.6 in [36]. As we proved in Lemma 2.5.1, 01?, => 0— with 0],", E A(l~/,H) C
TWO}, H) (see Corollary 2.4.1). Therefore 0‘ E Aw(17, H).

Theorem 2.5.1 Let X be an (R1) process and let Y, = Xt—fg DXsds be an (Rm)

process. Assume that 02 is regular. Then there exists a 2—cylindrical H ~martingale

76

W, such that for every h, g E H,
E{(Wt(h) — We(h))(Wt(g) — We(9))|}'s} = (t - 8)(h, 9);;

and

t ~
X, = / DXSds + (/ MW),
0

The above equality is in the following sense, ‘V’h E H, (X, — f; DXsds,h)H =
(f odW),(h) in M%(R). In particular, f odW is a 2-cylindrical H —martingale

associated with an ordinary H -valued martingale.

Proof. We define W = (f 0—d17)“’. Let us first prove that for X E A’”(i~/, H)

we have
E{[((/de/),—(/de/)',"M)[((/Xd17),—(/Xd1?)';”)(g)]|f,} (2.7)
= E{/( X; (h) ooX:(g),e2 M(r)>dr|.7~',} vn e H.

Recall that by condition (2) of Definition 2.4.6 and because Qy = (Iz/tro2 and
dallYll = tr02d(P (X) A), the expression

(X*(h) a X*(g), 02) = (X*(h) ® X*(g), oy) troz

is well defined on H X H.
We first obtain equality (2.7) for elementary processes of the form (2.3).

E{[(/Xdl7);”(h)—(/Xd17);”h)][(/Xd1~/),)—(/Xdl7),(g)]|J-‘,}
: EU: 1Ft((Yttaui(h))H _ (YStiui(h))H)l

i=k

Milena an» —(n.,u;(g))e)nf.}

77

where we assume (by refining the partition of I if necessary) that s), = s and tn 2 t.
The components with factors for which i yé j will give zero by the martingale

property of Y. For i = 3' we compute for each component

E(1Hu( :(h) on: (g) (Y. — mm) Ia}
= (1.,n,*(h)® *(g)E{/:o2( r)dr|r,,})
= E{/1p,x(,,,,,]u;‘(h) ®u,’-‘(g),02 (H)>de|r,.}.

By taking conditional expectation with respect to (F, C 7:3,. and summing all
terms from i = k to n we get the desired result.

Next,
E{/8t|(ooX;(h), ooX;(g ))H— (ooX*(h), ooX*(g ))Hm}
s (E(/ Hao( (X. —X> WWW (E{/ IIaoX*(e >IIHdA}>%

+ (E([ no o( (X. — X)*(e>IIHdX}> )%( (Etf Ila o X;:(h>IIHdX})%
g M:(X,, — X)N:(X) + N:(X,, — X)N:(X,,).

Therefore convergence Xn => X implies that, Vh E H,
t t
H] (mm o X;(g>.02> MA} —> E(/ (X*(h> o X*(g>, 0-2) dAlfs}

in L162) by contractivity of conditional expectation. Convergence Xn :> X implies

also that, Vh E H,
(/ X,dY)w(h) —> (/ XdY)w(h) in M§(R)
which, in turn, implies

E{[((/XdY),— (/X,,:ndY))( fXdY (fXdY), w)(g)]|J-'.}
—+E{[( (fXdY): —:(/XdY) h,)][((/XdY) —(/XdY): )(g)]|f.}

78

 

in L1(o), Vh e H. This concludes the proof of equality (2.7).
Using (2.7) we can now prove that
E{[Wt(h) -— W.(h>nWH(g> — moms}
= E{((/ Noah) — (f more»
x [(/0“d17)2”(9) — (/ a-dY>:”(g>uJ-'.}
= E{/: ((a->*(h) ® (a-)*(g), 02) was}

: E{ t(h,g)Hd/\|]:s} : (t — S)(hag)H

S

(for the third equality recall the proof of Lemma 2.4.2).
To show the last assertion of the theorem let us prove that a E [\(W, H) and

that for an elementary process Xn of the form (2.3),
(/ XndW) = (/ Xn o a'dT/Y" (2.8)
(implicitly Xn o 0— 6 Jim“), H)). Indeed,
2d~</ 2d~=/E 2 dA<.
f,“ HanHH) (aw! _ IIaIIHHH lawl , {IIaIIHHH} oo

Last equality follows from the fact that

E{E{1F(Wt(h)Wt(g) _ Ws(h)Ws(g)lfs}}

= P(F))\((s,t])(h,g)H

(h <59 sham/(F X (s,t]»

for F E .73, h,g E H and s S t, s,t 6 1. Therefore on = (P®A)tr as tr 6
(H (8)1 H)* is the extension of h (8) g H (h,g)H. Hence, lan = P (8) A.

Finiteness of f, E{||a||§,®lH}dA follows from the property (Rm) of Y. Also, a E
[\(W, H) (see the proof of Corollary 2.4.1).

79

Let {X,,}f,°=1 be a sequence of elementary processes. First we will establish that

X, o 0‘ E RWY, H). The domain D(X, o 0‘) is dense in H and

((X, 0 0—)”‘(hl ‘8 (X, 0 0—)*(9), Qy) = {7:163(IdD(a_) 0 X300, 1dv(e-) ° X;(9))H

extends uniquely to a continuous bilinear form (1 / tr02)(X,’:(h), X;(g)) H.

Further, Vh E H (using the above extension),

fox; ((X” o ”T(h) ® (X" 0 ”T(h)’ QY) dallYll =/ ||X3(h)||§;d(P (8 A) < oo.

9x1
Therefore, X, o 0‘ E I~H(Y, H) C Z‘”(Y, H).
Since 0‘ E AWY, H), there exists a sequence {Z,,,}f,‘,’___1 of elementary processes,

such that Z, => 0' as m —-> oo. Now, for every h E H,

fox/“X" 0 Zn)“ — (XH o a‘)*](h) s [(X, o z,)* — (X, o o-)*](h), QYWHYH

: {2x1 <(Zm — 0—)*(X':(h)) <8) (Zm _ 0—)*(X;(h))7 QY> danyn -> 0.

Since {Xn o Z,,,}‘,’,,°=1 is a sequence of elementary processes, X, o 0' 6 IV“ (Y, H).
Next, we will prove that if X, —> a in [VI/Y, H) then Xnoa‘ => IdH in [\WY, H)
as n ——> 00. Clearly Rig 6 1~\(Y, H) C AWY, H). Observe that
02
<(Xn o 0- — IdH)*(h) (8) (X, o 0— — IdH)*(g), —>

tra2

1

= tm,((1dp(,-, o x; — a)(h),(1do(H—) o X; — 0)(g))H

 

extends uniquely to a continuous bilinear form on H X H, namely, to

—1—((X:; — a>(h). (X; - a>(g)>H.

lira2

For every h E H, and for this extension, we have

[M ((XH 0 0‘ — IdH)*(h) (a (X, o o- — IdH)*(h), QY> dO‘IIYII

: Qx1<()('n — 0)*(h) (8) (X71 _ GYM), t7“) dlaVl/l —> 0

80

Since convergence in IV” implies convergence of stochastic integrals in MflR),

we conclude that Vh E H,
(j X, o zde)(h) —> (f X, o e-dY)w(h) as m —+ oo

and

(f X, o e-dY)w(h) —+ (/ IdeY)(h) = (Y, h)H as n —> oo

in MflR). Equality (2.8) can be proved as follows:

(/ XHdW)H(h)
= 2: 1H,.[(/ a-dY>::H(u:(h>) — (/ a-dY>::H(u:(h)>]
= Z 1H(M%(R) — ,gnwuf zdeHHXum» — (/ szY>..H(u:(h>)n

s
all
H

1H.[M%(R> — ,gggoK/ u. o szYHMh) — (f u.- o Zde>..H.(h>n

Ti

H
i

%(R> — 4320:1qu o zdeHHH-(h) — (f u.- o zde>..HH(h>l
= Wm) — 413,514] X. o zde)H(h>
= (f X, o e-dY);“(h).

Because, Vh E H,

~

(/ X,dW)(h) —> (/ adW)(h)
in MflR), we get
(X, — jot DXSds, h)H = (Y,,h)H = (/ odW),(h),

in MflR). This concludes the proof.

81

Let us mention that the assertion of the last theorem states in particular that
W is a 2—cylindrical standard Brownian motion, provided Wt(h) has continuous
sample paths in t for every h E H. The following Proposition gives some regularity
of the process W.

Proposition 2.5.1 Under the assumptions of Theorem 2.5.1, if X : I ——> H is

continuous, then Vh 6 H the real valued martingale

at

W) = (/ o’d?)‘”(h)
has P-a.e. continuous paths.

Proof. The proof is based on the following Lemma:

Lemma([35]). Let {M ”}f,°=1 C M? be a sequence of H -valued martingales which
converges in M? toward M. Then there exists a subsequence {M ”k},‘:°=1 with the
following property: for P—almost all to E Q, the paths t +—> M7”c (t,w) converge
uniformly on I to the paths t r—> M (t, w).

Since t I—-> DX, is continuous from I to L1(SZ,H) we can choose its jointly
measurable version in (t,w) (see [16], Theorem 1.2). Hence, t +—> f5 DXsds is
continuous from I to H. This, together with continuity of X : I —> H gives
continuity of Y : I —> H. Since for an elementary process of the form (2.3) the

stochastic process
(/ XHdY)(h> = :1 (mom) — have»)
has continuous sample paths then, by choosing X, => 0—, we get,
Vh e H (/ X,dY)(h) —> (/ o-dY)w(h)

in MflR), which completes the proof.

82

Chapter 3

Anticipative Stochastic

Differential Equations

3. 1 Introduction

Anticipative stochastic integration naturally leads to development of the theory
of anticipative Stochastic Differential Equations. This allows for analysis of non-
adapted processes as solutions of these equations. Anticipative SDE’s were con-
sidered by several authors. In particular Skorohod-type SDE’s were studied by
Buckdahn,Nualart Ocone and Pardoux ([5]-[8], [39],[40],[42]). Another approach
was presented by Ogawa, [46]-[47], where the author used his concept of stochastic
integration.

A natural way to obtain an anticipative SDE is to impose a boundary condition
to be a future-dependent random variable. In particular one can consider equations
with boundary condition of the type X0 = (p(X 1). An interesting presentation of
Ogawa-type SDE’s is given in [46]. As a nice example the author considers Go

and Return problem. Recently, in [47] Ogawa studied multidimensional stochastic

83

integral equations, which were linear of Fredholm-type. This seems to be a strong
application of anticipative calculus.

In this Chapter we consider a Gaussian process {X,, t E T} with arbitrary in-
dex set T and we study consequences of transformations of the index set T on the
Skorohod integral with respect to X. We obtain applications to time and space
reversal in case of Brownian motion and Brownian Sheet. Even though we con-
sider here general transformations of the parameter set our motivation came from
the Time Reversal Problem of a diffusion process and applications of this method
to the problem of filtering. In the case of Skorohod linear difiusions we obtain
existence and uniqueness of the solution for the reversed equation (a problem con-
sidered in [42]). As an example we formulate and solve Go and Return problem
for Skorohod linear diffusions. Further applications of anticipative stochastic cal-
culus and kinematics of Hilbert space valued stochastic motion to Time Reversal

Problem and Filtering Theory will be a subject of future research.

3.2 Skorohod Integral under Transformation of
a Parameter Set

Assume that {X,heT is a centered Gaussian process defined on a probability space
(52, .7, P) and indexed by an arbitrary parameter set T. The covariance function
of X will be denoted by C X and the RKHS of C X will be denoted by H (OX).

Definition 3.2.1 A map R : T —> T will be called a non-degenerate transforma-

tion of the parameter set T if
cl(span{Xt,t E T}) = cl(span{XR(,),t E T})
where ”cl” denotes closure in L2(Q,}', P).

84

For any transformation R : T —) T let T1 C T be such a set for which Vt E
T, T1 0 R"1(t) is a single element of T. Thus R : T1 —> R(T1) is a bijection. In
particular if R : T —> T is bijective then T1 = T. There are possibly many choices
of T1. Now we consider behavior of the Skorohod integral under non-degenerate

transformations.

Proposition 3.2.1 Let {X,}teT be a Gaussian process and R be a non-degenerate
transformation on T. Denote by I 3} the Skorohod integral with respect to X and by
I 5212, the Skorohod integral with respect to ( a Gaussian process) X R = {X R(,)}t€Tl.
Then:

(I) A map f, I—> ff 2 f(R(t1),...,R(tp))|Tlp is an isometry from H(CX)®P onto
H(C§%)-

(2) Ifu E ”D(IjQ) then uR = {uR(t),t 6 T1} 6 D( 32R) and

1;, (u) = 1;,H(uR). (3.1)

Moreover, denote by DX and DXR the Malliavin derivatives with respect to X

and X R respectively.
(3) If u, e D(DX), t 6 T1, then of e D(DX”) and
DjwulCt = D§(,)uR(t), s, t 6 T1, P-a.e. (3.2)
The equality is in the sense of H (0X12), with s 6 T1 as the variable. Also
Dsut E H(C'X)®2 (s,t E T) implies DfiwuiL2 E H(CxR)®2 (s,t 6 T1) and the
equality 0f norms, ||D3ut||L2(Q,H(CX)®2) = llDfRulillL2(Q,l-I(CXR)®2)'
va E L2(Q,H(CXR)) then,

(4) v = UR for some u E L2(X , H (Cx)) and llv||L2(n.H(cxH)) = IIUIlLe(n.H(cx))-
Moreover, v E D(ij) implies u E D(IjQ) and v, E D(DXR) implies u, E D(DX)
with D§(I:)v, = Dfut for s,t 6 T1.

85

In the case of Dvat E H(C'th)®2 (s,t 6 T1) also Dsu, E H(CX)®2 (s,t E T)

and the H—S norms of these derivatives are equal.

Proof. (1) Let us denote fR(t1,...,t,,) = f(R(t1),...,R(t,,)) for (t1,...,t,) E
Tf’. Thus ff(t1,...,t,,,t) = f,(R(t1,...,R(t,,),R(t)),(t1,...,tp,t) 6 T5“. Denote
H(X) = cl(span{Xt,t E T}) = cl(span{XtR,t 6 T1}). Let f(t) E H(CX). Then
f(t) = E(Xt7rx(f)) with 7rX(f) E H(X) and, for any t 6 T1,

fR(t) = f(R(t)) = E(XR(t)7rX(f)) = E(Xll7rx(f))

i.e. fR E H(CxR) and nXR(fR) = 7rX(f).
Also, if g E H(CXR) then, for t 6 T1,

 

g(t) = E(X.R«X”(g>) = E(XH(.)«XR(9>>.
But, nXR(g) E H(X), then f(t) = E(X,7TXR(g)) defines an element of H(CX).
Hence, g(t) = f(R(t)), t 6 T1 and
H
llgllexH) = HEX ||L2(n.f~.P) = llfllH(Cx)'

Now (1) follows for any p in view of the form of ONB and scalar product in the

tensor product of RKHS’s.

(2) In order to obtain (2) we will first prove that for every p the following equality
holds:

[g((fp) : fling?)- (3-3)

Note that we have already proved the above for p = 1, as I1 = it by definition.
For p = O equality (3.3) is obvious.

Every f, E H (C X)®P can be represented as a following series (see Section 1.3.1):

f(t17t21”')tp) : Z aa1,a2 ..... ap eal (t1)€a2 (t2)...€ap (tp)

(11,02 1111 a]!

86

with Za1,a2,...,ap aghazmap < 00 and {emoz = 1,2, ...} an ONB of H(C'X). Then
ff 6 H (Cxfl)®p by (1). It is enough to prove equation (3.3) for functions of the
form: eal (t1)e,,2 (t2)...eap (tp), because for arbitrary f, E H (CX) we will have

n1 “P ~

[g((fp) : limn1,...,np—>OOI§((( Z Z aal,...,ozpea1---eozp))

a1=1 ap=1

_ R ~
— lzmnl ..... np—mo Ixfl( (:m 2? aa11' -eaap 011' R'”eap))

a1=1 ap=1

: IPR(lzmm ..... ’np—)OO (:-- n: aal ..... ape a1“ ' eap))'

a1=1 op=1
We used properties (5) and (6) of Multiple Wiener Integrals (see Section 1.3.1)
as well as a simple fact that the operations f H fR and symmetrization ”"”
commute.

In View of (1) we have, 20,, a, a, aalm, , eR 6R” .eR —> ff in H(CxR)®p and

“p 011 02 Op

hence (2am, ,,,,, a? Claim... ,,e§,e§2.. .efp) —> (ff) in H(CxR)®p. Thus,
152a.) = Inuit?) = 12120,?)-

Let us now prove equation (3.3) for functions of the form ea,1 (t1)ea,(t2)...eap(tp)
for p > 1. We can use property (8) of MWI and we only need to show that for
(t1, ...,tk_1, tk+1, ...,tp) E Tlpnl, one gets

R
[(prEgl)XlR(tla "')tk—13tk+17°"7tp) : (fflfgfz)x (t1) "'atk—1atk+la "')tp)a

Where the superscripts X and X R outside the brackets indicate that the operation
”g” is taken in H (C'X) and H (C'Xn) respectively.
We have

(fp%gl)x(t1, ..., tk_1, tk+1, ..., tp) :

= (fp(t1, ...,tk, ...,tp),g1(tk))H(Cx)

87

: E{ITX(fp(t1, ---)tk—11'atk+17 ...,tp))’/TX(91())}
2': E{7I'XR(f;z(k)(tla “'3 tic—1) '3 tk+13 "'1 tp))7rXR(in())}
: (f:(k)(t13“'atk)-“)tp)igf(tk))H(CXR)

R
: (f:(k)%gf)x (t1) "'7 tk—l) tk+19 "'3 tp)

where R(k) transforms only the kth coordinate with t1, ...,tk_1,tk+1, ...,tp fixed.

But the above implies that

l(fp%91)XlR(t1) “'3 tic—1) tk+1a ---a tp)
: (fp®gl)X(R(tl) B(tk—1)iR(tk+1)a "°)R(tp))
=(fR<R>eg Rt)R”(E( t1...) H(tH-H).R(tH.H). ....R(tH))

: (fffg: )XR (t1, ..., tic-1) tk+1a "-3 tP)‘
Thus,
13((fp§91)")=13R(l(fp§91)X]R)HfflngERVR)

which allows us to use the inductive relation (8) for Multiple Wiener Integrals to
complete the proof of equality (3.3).
Now if u E ’D(I§() and u, 2 22:0 Ip(fp(t1, ...,t,, t)) then, for t 6 T1,

HHH) = i 1,? (fH(-.E(t))) = iIR’RoR-s»

p=0
hence,
1:.(u) = 21.55.1(fp)=2135.’i((ip)R )
p=0 p=0
= 2am R))=I§.n(uR)

proving (2).

88

 

(3) Let u E D(DX). Then, for s,t 6 T1,

00

DR(s)uR(t) : Zplgi:1(fp('vR(3)vR(t)))

p=1

= 2121,5512de st))=D§RU?-

The equality of norms claimed in (3) follows by Lemma 1.3.2, (2) and by (1) of

this Proposition.

(4) To prove (4), let v E L2(X, H(CXR)). Then for t 6 T1 we have

= i1:“(gH(-.t)) = Emma),

as by (1), for any g E H(CxR)®(p+1) there exists f E H(CX)®(”+1) with g = fR.

Hence, for t 6 T1,

___le&< fR(' =ZIX( ((f, t)))=uE(t)-
P=0 p=0
According to (1),u 2:10 If (fp(-, t)) E L2(X, H (CX)) and equality of norms

claimed in (4) is satisfied. The last part of assertion (4) follows from (1),(2) and (3)
since failure to satisfy any stated condition by u implies violation of this condition

by v.

Example 3.2.1 Transformations of parameter set and Skorohod integral.

1. Brownian motion and Time Reversal. Let .77, 2: o{B,, s g t} and {ut,t 6
[0,1]} be (T}),EmJ] adapted stochastic process, such that u E L2(Q,L2[O,1]). Then
{R = B1 — B1_t,t 6 [0,1]} is also a Brownian motion and {at = u1_,,t 6 [0,1]}
is adapted to filtration .73 = o{B1 — B3,t g s S 1}. Denote B, = B1_t. We have

fol UtdBt = [h(fo. urdr) = ISB(‘/1—iu,.dr). (3.4)

0

89

By the same method as in the proof of Theorem 3.2.1 we can show that

~

I;((/,'u.d1~)) = Inf, ear)

with (f6 urdr) 2 f01 urdr — f01—' urdr. Hence, we get,

[01 11,113, = Ig((/O' 11.11155) = [gt—1):]: a, * dB,
where I i is the Skorohod integral defined in [41] (see Example 1.3.2) and ” * ”
denotes the backward Ité integral. We obtained the relation: I E; (u) = I}, (27.) given
in [42]. In particular a E D(Ig). Note that in [42], the process 31-, - Bl = —Bt
was used as an integrator. But it is true that I 3 = —If_X), which is easy to check
using recursive properties of Multiple Wiener Integrals.
Note also that B, is not a Brownian motion process and the Equation 3.4 is

reversed pathwise in H. In the case of Brownian motion, we also have,

at

1-. .
[g(f usds) = 183“] u,ds)).
o 0
Indeed,
1—- - . . -
I§(/ usds) 2 I133(/ usds) = I};(u) : [g(a) = I§(/ u1_3ds)
0 o o
1 1—-
= I§(/ usds ——/ usds).
0 o
2. Brownian Sheet. Let T = [0,1]2 and let us think of a point (:13, t) E Tas the
space-time parameter. Let W(a:, t) be a Brownian Sheet ([58]), that is, a Gaussian
process {Wt,t E T} defined by covariance CW((33,t), (y, s)) = (a; /\ y)(t /\ s), i.e.
CW 2 CB (8) CB where CB is the covariance of Brownian motion. In this case we
also have that H (CW), the RKHS of Cw, is the tensor product of the RKHS’s
H(CB) Of 032 H(Cw) = H(CB)®2.
(a) Time Reversal. Let R(:I:,t) 2 (113,1 —- t), then,

MHQWC’EH t)) = 15m,1_1)(U($11 — t))-

90

 

(b) Space Reversal. This is the case of R(:z:, t) = (1 — 3:, t).
3. Generalized Processes (see Example 1.2.1 (0) and [19]). Let T = 03°(R)
and consider Generalized Wiener Process {BW (p E T} given by covariance func—
tion C((p,i/2) = f (a: /\ y)cp(ac)z/J(y)d:cdy. Consider R : T —> T a non—singular
transformation of the form: R((p)(:r) = (p(r(:2:)) E T. Let {HoloeT E D(Ig) then
{Hither 6 D(IER) and JEROME) = H“)-

In the particular case of R((p)(t) = (p(—t), R is non-singular and we have the

following ”time reversal”:

21M) = Ig,,_,(u..1_.)) = 1:1,, (now) = 12(1)).

4. Ogawa Line Integral Let {X,, t E T} be a Gaussian process and 7 : S —> T

be a bijective parametrization. Let Y, 2 X7“), then

(i) CX(7(81)17(82)) = (LE/(81182)

(ii) H (OX) and H (Cy) are isometric under the mapping f )—> f o 7 E H (Cy) for

f E H(Cxl-

(iii) 7rx(f) = 7rY(f 0’7) for f E H(Cx)-

Thus, 6°(u) = 6°(v), for v, = u,(,), provided either of the integrals exists.
Consider the Brownian Sheet {W(,,,t), (93,t) E [0,1]2}. One can define Ogawa

line integral, I‘ — 6", over a curve P C [0, 1]2 with respect to {W(:v,t)) (3:, t) E F} in

a usual way. Now assume that I‘ can be parametrized by a function '7 : [a, b] ——> F,

0 g a S b S 1 and 7(3) 2 (71(3), 72(3)) with both coordinates non-decreasing and
such that the map

Y‘1(71(8).72(8)) = 71(8)72(8)
is bijective from I‘ to S' = [71(a)ryz(a),'yl(b)'72(b)]. Then :5, : S ——> I‘ is a bijective

parametrization and the process B, 2 WW.) is a Brownian motion. Hence,
I‘ — 63,,(u) = 633(1)) : f1), 0 dB,
5

91

where v, = us“) and the last integral is in the sense of Fisk and Stratonovich (see
[45] for a definition) and is assumed to exist. In particular if u(x,” = f (W(,,,t)) with

f E 0'2 then

r — 6w(f’(W)) = /, f’(BH) o dB. = f(W(H.(b).HH(b)) — f(W(’Y1(a), 1201)).

Thus in this case, the Ogawa line integral shares the property of the Lebesgue
integral. Properties of the Ogawa line integral and its relation to line integrals of

Cairoli and Walsh [9] will be a subject of further investigation.

3.3 Skorohod-Type Linear Stochastic Differen-
tial Equations

The class of Skorohod Linear SDE’s was considered by Buckdahn in [6] where the
author proved existence and uniqueness of the solution. We give a short review of
this result.

Assume that { Bt, t E [0, 1]} is a Brownian motion defined on a probability space
((2, J", P). Here (I = 00([O, 1]). We consider the following Skorohod Linear SDE:

t .
Z, = 17 +/ b(s)Z(s)ds + 11(021[O,]), 0 g t g 1 (3.5)
0

where b E L2([0,1],Loo(fl)),17 E Loo(fl)),o 6 141.00 = L2([O,1],D1’°°). The space
D1’°° is defined as follows. Let

8 : {F : f(Bt1)°")Btn)7n Z 13t17 "'atn E [011l7f E 050(Rn)}’

where C§°(Rn) denotes the space of C°° functions which are bounded with all
their derivatives. Recall the Malliavin derivative Di of [41] (see Example 1.3.1).
Denote by Dl’2 the closure of S in the following norm: llFll1,2 2 [IF [I L252) +

92

||DiF || L2(91L2([0,ll))' Then Dl’°° is the restriction of D1:2 to these random variables
for which llFll1,oo = HF“,o + II [IDIFIIL2([0,1])”oo < 00. The stochastic integral If is
again as in Example 1.3.2.

The main result on Skorohod Linear SDE’s in [6] is the following.

Theorem 3.3.1 Suppose a E L1,oo,b E L2([O,1],LOO(Q)),77 E LOO(S2). Denote by

{Tt,t E T} the family of transformations associated with o as follows:
tA-
Ttw = w +/ 03(T3w)ds, w E C0([0,1]).
0

Let A, be the inverse to T, and L,, the density dP o Tfl/dP, where P is the
Wiener measure on C'0[O,1]. Then the process X defined by

X, = 77(At)ea;p{/ot b,(T3At)ds}Lt, t E [0, 1] (3.6)

belongs to L1([0, 1] X 0), 0X 1l01tl 6 DH“) Vt E [0, 1] and it verifies equation ( 3. 5 )
Conversely, ifY E L1([0, 1] X (2) is such that oYlw] 6 DH“) Vt E [0, 1] and
verifies equation (3.5) and if moreover o,b E Loo([0, 1] X It) and the Malliavin
derivative Dio E L<,<,([0,1]2 X 9), then Y, is of the form (3. 6) Vt 6 [0,1].

Our purpose is to reverse equation (3.5). We begin with a supporting Lemma.

Lemma 3.3.1 Let {u,}s€[0,l] be such that u,1[0,t](s) E D(Ifg) Vt 6 [0,1]. Then
for the time reversed process, a, = u1_3, we have u31[0,t](s) E D(Ig) Vt E [0, 1]

and if we denote X, = I];(1[0,t](s)us) then,
X1_t — X1 = —IiB(1[0,t](S)fi3).

Here, B, 2 Bl — Bl_t.

93

Proof. Because for v, E D(Ig) we know that o, E D(Ig) (as pointed out in
Example 3.2.1), by linearity of the domain of the Skorohod integral, we conclude
that

XI—t — X1 = [h(—1[1-t,1](3)us) = —Il§(1[1—t.1](1 — S)u1—s)
= —I;'~,(1[0,](s)1-1,).
In particular 1[0,,](s)u, E D(Ig).
Cl

Now we can easily derive a result about time reversal for Linear Skorohod SDE’s.

Theorem 3.3.2 Assume that the coefi‘icients of the linear Skorohod SDE satisfy
assumptions of Theorem 3.3.1. If {Zt},€[0,1] is the solution of equation (3. 5) then
the time reversed process Z, = Zl_t, is the unique solution in L1([O, 1] X 9) of the

time reversed equation
_ t _ .
X, — z, = [0 —b(s)X,ds + 1;,(—1[0,,15X) (3.7)
where b(t) = b(1 —— t), ("f(t) = 0(1 — t) and B, = 31 — Bl_,.

Proof. We need to prove uniqueness only. Let Y, E L1([0, 1] X 9) be another
solution of the Equation (3.7). Then Zt—Yt E L1([0, 1] X 82) is also a solution of (3.7)
with vanishing initial condition. Now all the assumptions of the Theorem 3.3.1 are

satisfied. Hence Z, — Y, = O.

94

Example 3.3.1 Linear difiusions.

Applying Theorem 3.3.2 to the following diffusion equation:
X,=:E0+/0b( (s),de+/0( 5),,de
we get
_ _ t _ _ t _ ..
X, = X0 + / —b(e)X,de + / —c‘r(s)X, * dB,
0 o

where the last integral is the backward It6 integral. Thus we obtain the result in

[42] in this special case of linear equation.
Example 3.3.2 ”Go and Return” problem.

Let X (t, 3:), Y(t, y) be the unique solutions of the following equations:

X(t,:c)=3:+/Otb(s)X (,s3:)ds+/Oo(s (,)sa: )0st (G)

Y(ty)=y— [1(8)(8y)d8-/0(8)Y(8y)0d33 (R)

t
The solutions X,, Y, are adapted to o{B,,, s S t} and o{Bl—B,, s 2 t} respectively.
Ogawa ([46]) proves that Y(t,X (1,511)) solves a modified Equation (R) with y
replaced with X (1, 3:) and the stochastic integral changed to the Ogawa integral

6‘}, with respect to the system of Haar functions. Moreover the following equality

holds P—a.e., V3: 6 R,
Y(O, X(1, 33)) = 3: (G — R)

Note that the described above ”Go and Return” problem is meaningless, unless
it is stated with help of anticipative calculus.
Let us now consider the ”Go and Return” problem in terms of It6 and Sko-

rohod SDE’s. Since the rules of integration here are different from those for the

95

 

Stratonovich and Ogawa integrals, one cannot expect that Y(t, X (1, 33)) will be a
solution for the Skorohod equation corresponding to the Equation (R) if X, is a
solution of the Ité or Skorohod equation corresponding to the Equation (G). Also
the ”Go and Return” relation (G-R) may be violated. Indeed, let us examine the

following example:

t
X, = H + / XSdB, (GI)
o
1
n=y—fde. (R’)
t
The solutions are given by
l
X, = 33exp{B, — —2-t}
1
Y; = yexp{—(E1 — Bt) — ,(1 — t)}
In this case we have, Y(O, X(1, 33)) = 1136—1 and Y(t, X(1,3:)) does not satisfy
Y(t,X(1, 11)) = X(1,3:) — IR(1[,,](e)Y(e,X(1, 27)» (RR)

which is easy to check by simply comparing expectations of both sides.
Because of the above example we state the ”Go and Return” problem for Sko-

rohod equations in the following way:

X(t. 21) = a: + f; b(s)X(s. sods + 1,(1,,,,(,)0(.)X(., 22)) (0.,

Y(t,X(1,m)) = X(1,3:) —/t1b(s)Y(s,X(l,33))ds
— 12(11.,11(s)o(s)Y(s, X(1, a») (RS)
Y(O,X(1,33)) 2 3: (G — R3)

where the first equation is either an R6 or a Skorohod equation and we impose

conditions on the coefficients b and 0 sufficient for uniqueness and existence of

96

 

solutions for the Equation (GS). Clearly
it .
X(t,:c) — X(1, 33) = —/ b(s)X(s, 33)ds —— Ig(1[,,1](s)o(s)X(s,:c)).
1

Thus Y(t, X(1,33)) = X(t, :c) satisfies Equations (R3) and (G — Rs). Note that
in the case when (G5) is an It6 equation, this solution is adapted to the natural
filtration of Brownian motion.

Let us now consider the following equation:

Y(t,X(1,3:)) = X(1,33) — fotb(1 — s)Y(s,X(1,3:))ds
— Il§(1[0.t](3)0(1 — 3)Y(8,X(1a$))- (Ri)

If process X, describes ”motion” of a particle then process Y, can serve as a model
for motion of a particle with reversed ”velocity” b and under reversed random
forces 6B (see Chapter 2 for more detailed discussion of kinematic properties of a
random motion).

By Theorem 3.3.2 X (t, 3:) satisfies

X(t, 33) = X(1,33) — [Qt b(s)X(s, 3;)ds — Ig(1[0,,](s)5(s)X(s,3:).

Hence Y(t,X(1,:c)) = X(t,3:) solves the Equation (Rf), Y(0,X(1,3:)) = X(1,3:),
and

Y(1,X(1,3:)) =33. (G—Ri)
Moreover, under the smoothness assumptions of Theorem 3.4 on b and a, process
X (t, 3:) is the unique solution in L1([0, 1] X 9) of the Equation (Rf).
Finally, let us note that equations (Rf) and (R3) are equivalent,
Y(t) X(la 37)) _ X(1, it)

= _ l1b(S)Y(S’X(1’””dS — Ig(1[t,11(s)a(8)Y(8.X (1,8)»

97

= —/01—tb(1 — s)Y(1— s,X(1,3:))ds — Ig(1[,,1](1 — s)0(1 - S)
X Y(l — s,X(1,3:))),

which is equivalent to
Y(t, X(1,:r)) = X(1,H) — fOtE(8)Y(81X(118))d8 — 1R,(1,,,(e)o(e)Y(e,X(1, 3:))).

Thus also the Equation (R3) have a unique solution in L1([0, 1] X (2) since

otherwise the Equation (Rf) would have different solutions.

98

Appendix

A Abstract Wiener Space

Our framework concerned a structure related to general Gaussian process, i.e.
(i, H, X) with X a LCTVS. However, it seems desirable to review construction of
an Abstract Wiener Space (AWS) in order to have better general background and
understanding in the case when the Gaussian process is a Brownian motion.

Here, H always denotes a Hilbert space and B a Banach space. Both spaces
are real and separable. H * and B * denote the dual spaces. We will always identify
H "‘ with H.

A subset C of a Banach space B is called a cylinder if it can be written in the

following form:

C = {e E B: ((e,y1), ..., (e,y,,)) E A}

where {y1, ..., y,} C B* and A is a Borel subset of R”. By Cyl (B) we will denote
the collection of all cylinders in B.

Note that, for example, any C 6 Cyl (H) can be written in the form: C = {h E
H, Ph 6 A}, where P E P(H) (a finite dimensional projection on H), and A is a
Borel subset of P(H).

The (canonical) Gauss measure 7,, with parameter a > 0, on a Hilbert space

99

H, is a function on cylinder subsets of H defined by

7,,(C) = (27r02)'%/ e3:p{—|—|£EI—}d33
A 202 ’

for C E Cyl(H), C = {h E H: Ph 6 A}, P E P(H). Here n = dimP(H), || - ”H
is the norm in H and dx denotes the Lebesgue measure on P(H). We will write 7
for '71.

The Gauss measure ,7 has no a-additive extension from Cyl (H) unless the
Hilbert space H is finite dimensional (see [27] for the proof). To obtain a 0-
additive extension one constructs a Banach space B containing H and studies
o-fields on B.

A seminorm [I - H on a Hilbert space H is called a measurable seminorm if
Vs > 0 3P0 E P(H) VP _1_ P0, P E P(H): ”)((IlPhH > e) < e.

Let H - ll be a measurable norm on a Hilbert space H and define B to be a
completion of H with respect to the norm I] ~ [I. Then B is a Banach space and

the following relation holds ([21]):
B*‘-,>H'—+B
i i

where i is the natural embedding and i* is its conjugate: 2*(e*)(h) = e*(i(h)), with
e* E B *, h E H. Both embeddings, i and 2* are continuous and have dense ranges.
Let a, be a function on cylinder subsets of B, induced by the Gauss measure

on H, that is,

MAC) = “MC 0 H),
for any C 6 Cyl (B) We will also write ,u for #1. Note that the above definition of
,u, is correct, since B*;;>H.

Next theorem ([21]) provides the o-additive extension of R), the following (The-
orem 4.2, [27]) identifies the cylindrical o-field with the Borel o-field on B.

100

Theorem A.1 The set function ,u defined on Cyl(B) induced by the Gauss mea-
sure 7 on H has a o-additive extension {denoted further also by ,u) to the o—field
generated by Cyl(B).

Theorem A.2 The o-field generated by Cyl (B) is the Borel o-field of the Banach
space B.

The triple (i, H, B) is called an Abstract Wiener Space. The measure u on
B of Theorem A.1 is called the Wiener measure. The measure no, a o—additive
extension of the set function a, from Cyl(B), is called the Wiener measure with

variance 0”.
Example A.1 Standard AWS.

Let Co = CO[O, 1] be the Banach space of continuous functions on [0, 1] vanishing at
zero, endowed with the supremum norm. Let C" be the Hilbert space of absolutely
continuous functions in Co with square integrable derivatives, with respect to the
scalar product (f, g) = fol f’ (t)g’ (t)dt. Then the triple (i,C’,C0) is an AWS (see
[27] for the proof).

Now recall the Brownian motion process of Example 1.2.1 (a). There exists a
version of this process with continuous sample paths (Theorem 37.1, [4]). This
means that the Banach space C0 can be considered as the set of sample paths of
the continuous version of Brownian motion. As mentioned in Example 1.2.1 (a),
C’ is the RKHS of this process. The Wiener measure a on Co is the extension
of the Gauss measure on C’. For any C E Cyl(Co) of the form: C = {B E Co :
(B(tl), B(tz), ..., B(t,)) E A}, A - a Borel subset of R”, the Wiener measure u(C)

can be expressed as follows:

101

 

 

NIH

u(C) : [(27r)"t1(t2 —- t1)...(t,, — t,_1)]—

2 2 2
u1 (u2 — ul) (u, — u,_1)
X _ Ill 2 d IOOd n-
fAemfl [t1+ 252—111 + + t,—t,,_1 ]/}11, u

 

This is the probability of the event {(3 (t1), B (t2), ..., B (t,)) E A} for Brownian
motion B = {B,, t 6 [0,1]}.

B Backward It6 and Fisk—Stratonovich Integrals

With Brownian motion {B,,t E [0, 1]}, we associate two filtrations, .77, = o{B,,, s S

 

t} and .7:1 = 0‘{31 — BS, 3 Z t}, with the convention .70 = .771 = the trivial 0—
field. The first filtration is increasing as t increases and the second is increasing
as t decreases. The forward It6 integral is defined for processes adapted to the
natural filtration, {f,},€[0,1]. The Backward Ité integral can be defined for pro-
cesses {u,,t 6 [0,1]}, adapted to the “backward” filtration {F},€[0,1] , satisfying
condition E /0 1ugds < 00.

Let u), E L2(Q,FR), k = O,1,...,n, un+1 an .71 measurable random variable.

Assume that the sum
71,—]

Z uk+llltt,tk+1) + “n+11m
k=0

converges to u in L2(Q X [0,1]). Here, 0 = to < t1 < t2 < < t, = 1. Then, the
backward It6 integral of u is defined as the limit in L2(Q) of the sum

n—l

Z Ult+1(Bt,,+1 '— Btk)

k=0

1

and denoted by f u, * dB,. This definition is not ambiguous because the limit
0

defining the backward It6 integral does not depend on the choice of the sequence

approximating u in L2(Q X [0,1]).

102

Note, that the backward It6 integral of u coincides with the forward It6 integral
of the process {17, = u1_,,t 6 [0,1]} with respect to Brownian motion {31 —
B1_,, t 6 [0,1]}. For more extensive discussion and applications we refer to the
work of Kunita [26].

A random process {u,,t 6 [0,1]}, such that P(/01u,2dt < oo) = 1 is said to be

Fisk—Stratonovich integrable, if the following limit:

 

. "—10%. —Ut)
"11%; ”+12 k (Btk+1 — Btk)

exists in probability for any sequence of partitions O = to < t1 < t2 < < t, = 1,
with max{tk+1 — tk, k = 0, ...,n — 1} —+ O as n —) 00 and the limit is independent
of the choice of the sequence of partitions. The Fisk—Stratonovich integral of u is

1
denoted by f u, 0 dB,. For further properties and references see [45].
0

C Hilbert—Schmidt and Trace Class Operators
on Hilbert Space

Let H be a separable Hilbert space. A linear operator T : H —> H is called
Hilbert—Schmidt if it admits a representation of the form

00

Th 2 Z A'n,(h’) hn)Hen

n=1
where h E H, {e,}f,°=1, {h,}:°=1, are orthonormal sets in H, A, > O, n = 1,2,
and 2?le A3, < oo.

Equivalently, T is a Hilbert—Schmidt operator on H if for some (hence for any)
orthonormal basis {e,},‘i°___1 C H, illThnlliq < oo.

n=1

103

With the above notation, the Hilbert—Schmidt norm of a Hilbert—Schmidt op-

erator T is defined as follows:
”Tllz =(ZIIThHIIi1)R =(Z A3,)8.
n=1 n=1

The middle sum above is independent of the choice of the orthonormal basis.

The collection of Hilbert—Schmidt operators on H, with the norm || [[2 is a
Hilbert space, denoted here by H ‘82. The scalar product of two Hilbert—Schmidt
operators T, S E H ‘82 is given explicitly by (T, S) H912 = i(Te,,Se,)H, where
{e,},’,°=1 C H is an orthonormal basis. ”:1

A linear operator T : H —) H is called trace class if

”T“, = sup 5: [(Th,, e,)H| < 00,

where the supremum is taken over all orthonormal systems of vectors {e,}f,°=1,
{h,};’,°=1 C H. The quantity ||T||1 is called the trace class norm of T.

Trace class operators on H, with the trace class norm [I [[1, form a Banach
space.

Every trace class operator is automatically a Hilbert—Schmidt operator and the

following relation holds:

l|T||1 Z ||T||2 2 ”TH

where the latter norm is the operator (supremum) norm.

In the conclusion, let us recall the notion of a tensor product of unitary spaces
(i.e. linear spaces with scalar products). Let H, K be unitary spaces with bases
{6,},E1, {jg-be] respectively. The tensor product H (8) K of the spaces H and K
is the linear space, whose basis is formed by the pairs (e,, f,), denoted by e,- ® fj.
With every pair 3: = 20361; 6 H, y = Zfijfj E K, we associate an element
x®y=Za,C,-e,®fj E H®K.

104

The linear space H (8) K can be made into a unitary space by defining the scalar

product as follows:

(1131 ® 9111132 ‘3 y2)H®K = ($1,$2)H(3/1, 312%.

In particular, H ‘82, the space of Hilbert—Schmidt operators on H, is the com-
pletion of the unitary space H (X) H in the Hilbert—Schmidt norm under the iden-
tification of Section 2.2.

The role of tensor product is emphasized by the fact that there exists a bilinear
map (,0 : H X K —> H (8 K, such that given any linear space L and a bilinear map
b : H X K ——> L, there exists a linear map l : H (X) K —+ L, replacing b, in the sense,
that b = l o (,0.

105

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112

 

 

 

   

 

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