The extended stochastic integral in linear spaces with differentiable measures and related topics

264 Pages·1996·80.335 MB·English

by  Norin N.V.

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THE EXTENDED
STOCHASTIC INTEGRAL
IN LINEAR SPACES
WITH DIFFERENTIABLE
MEASURES AND
RELATED TOPICS
Nicolai Victorovich Norin
Moscow Institute of Radio Engineering, Electronics and Automation
Moscow, Russia
V fe World Scientific
wik SSiinnggaappoorree* * NNeeww JJeerrsseeyy *•L L ondon • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Norin, N. V. (Nicolai Victorovich), 1955-
The extended stochastic integral in linear
spaces with differentiable measures and related topics / N. V. Norin.
272 p., 21.5 cm.
Includes bibliographical references (pp. 245-253) and index.
ISBN 9810225687 (alk. paper)
1. Stochastic integrals. 2. Function spaces.
I. Norin, N. V. II. Title
QA274.22.N67 1996
519.2-dc20 96-16453
crp
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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This book is printed on acid-free paper.
Printed in Singapore by Uto-Print
To my father
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Preface
The subject of this book is the extended stochastic integral (ESI) (this integral is often
called Skorokhod-Hitsuda integral) and its relation to the logarithmic derivative of
a differentiable measure along a vector or an operator field. Besides, the theory of
surface measures and the theory of heat potentials in infinite-dimensional spaces are
discussed. These theories are closely related to ESI.
The book starts with a short account of classic stochastic analysis in the
Wiener space.
The remainder of the book may be separated into three parts.
The first part is devoted to the detailed discussion of the ESI for the Wiener
measure including properties of this integral understood as a process. Besides, the
ESI with a nonrandom kernel is investigated.
The second part is devoted to the definition and the investigation of properties
of the ESI for Gaussian and differentiable measures. Firstly, the vector ESI is defined
in the case of a Gaussian measure on a linear space; it is an analog of the Ito's integral
with a variable upper limit and it is a bounded operator in the scale of Sobolev spaces
of the vector-valued Gaussian functionals. The existence and uniqueness theorems for
the solution of the linear integral stochastic equation with this integral are proved.
Then the ESI is defined in the case of a differentiable measure as the
logarithmic derivative of this measure along a vector or operator field. The case
of the mixture of Gaussian measures is investigated in detail. In particular, the linear
integral stochastic equation with the ESI is discussed.
The third part of the book is about surface measures in Banach spaces and
heat potentials theory in the Hilbert space. It is rather non-linear functional analysis
but this part of the book is closely related to its other parts since the heat potential
of the double layer may be considered as the value of the logarithmic derivative of a
Gaussian measure along the specific vector field, which is conormal to the sphere.
Almost all results included in the book are not yet published in book form.
Moreover, these results were published in Soviet journals and may be unknown to the
western reader.
Chapter 1 deals with well known results on stochastic integration in the Wiener
space. The setting in this chapter is more functional-analytic than usual.
Chapter 2 is based mostly on the papers of M. Hitsuda, A. V. Skorokhod,
Yu. L. Dalecky, A. Shevlyakov, D. Nualart, E. Pardoux and many other researchers.
As far as I know these results were not published in book form.
Chapter 3 contains the author's results.
Chapter 4 collects the main results on differentiable measures. These results
vn
Preface
Vlll
were mostly obtained by Moscow mathematicians participants of the seminar on
infinite-dimensional analysis in the Moscow State University. This seminar was
founded by Prof. S. V. Fomin (he was the first one who gave the definition of a
differentiable measure). Now the leader of the seminar is Prof. 0. G. Smolyanov.
The results of Chapter 5 are taken partly from the book of S. Watanabe
"Lectures on stochastic differential equations and Malliavin calculus", and the rest of
the results belongs to the author.
The main results of Chapters 6, 7 and 9 belong to the author.
Chapter 8 is based on the works of the Russian mathematician A. V. Uglanov
with some improvements made by the author.
I would like to express my gratitude to 0. G. Smolyanov for introducing me
to the world of infinite-dimensional analysis, his constant interest and encouragement
during the writing of this book. I am also indebted to A. N. Shiryaev for his
support of my investigations, to Yu. L. Dalecky, A. V. Skorokhod, E. T. Shavgulidze,
A. V. Uglanov, M. Hitsuda, S. Watanabe, A. A. Belyaev, V.-K. Bentkus, A. A. Doro-
govtsev, Yu. M. Kabanov, A. I. Kirillov for the numerous discussions concerning the
subject of this book. Special thanks to A. Yu. Veretennikov whose support was very
important for me. I am also grateful to S. K. Lando for his careful reading of the
manuscript and polish the English. Finally, I would like to thank Ms. E. H. Chionh
and the staff at World Scientific who provided much help in the publication of this
book.
Contents
Preface vii
1 Stochastic calculus in Wiener space 1
1.1 Basic notions. Ito integral in C[0,1] 1
1.1.1 Multiple stochastic integral 6
1.1.2 Hermite polynomials and multiple stochastic integrals 8
1.2 Ito's formula for stochastic integral 13
1.3 Bibliographical remarks 13
2 Extended stochastic integral in Wiener space 15
2.1 Basic definitions 15
2.2 Local properties of stochastic derivative and ESI 25
2.3 Extended stochastic integral with variable upper limit . 28
2.4 Ito's formula for the extended stochastic integral 30
2.5 Bibliographical remarks . 31
3 Randomized extended stochastic integrals with jumps 33
3.1 Introduction 33
3.2 Randomized ESI and its continuity 33
3.3 Quadratic variation 37
3.4 Ito's formula 47
3.4.1 Ito's formula for randomized ESI without jumps 47
3.4.2 Ito's formula for jump discontinuous randomized ESI 55
3.4.3 Special cases of Ito's formula and some examples 57
3.4.4 Multidimensional Ito's formula 59
3.4.5 The probabilistic solution of some integro-differential
equation . . 60
3.5 Formula for Brownian partial derivatives 61
3.6 Bibliographical remarks 64
4 Introduction to the theory of differentiable measures 65
4.1 Measures in locally convex spaces 65
4.2 Definition of differentiable measure 67
4.3 Differentiability of Gaussian measure 78
4.4 The extended stochastic integral for the Wiener measure as the
differentiable measure . 83
4.5 Bibliographical remarks 85
IX
X Contents
5 Extended vector stochastic integral in Sobolev spaces
of Wiener functionals 87
5.1 Introduction 87
5.2 Ito-Wiener decomposition for Gaussian measure 87
5.3 Sobolev spaces of Wiener functionals ■ 93
5.4 The vector extended stochastic integral 96
5.5 The ESI in Hilbert space of generalized functions 106
5.6 Quadratic variation of extended stochastic integral in abstract Wiener
space . . 114
5.7 Bibliographical remarks 120
6 Stochastic integrals and differentiable measures 121
6.1 Introduction . 121
6.2 Main results 121
6.3 Applications 129
6.3.1 The image of differentiable measure 129
6.3.2 "Gauss-Ostrogradsky" formula. 130
6.4 Some formulas for differential forms of finite codegree 131
6.5 Bibliographical remarks 136
7 Differential properties of mixtures of Gaussian measures 137
7.1 Introduction 137
7.2 Mixtures of Gaussian measures 138
7.3 The direct integral of Fock spaces 140
7.4 Ito-Wiener decomposition for mixtures of Gaussian measures . 141
7.5 Logarithmic derivatives of the mixtures of Gaussian measures 145
7.6 Extended stochastic integral for mixtures of Gaussian measures 147
7.7 Ornstein-Uhlenbeck semi-group 151
7.8 Commutation relations . . .. 152
7.9 The ESI in Hilbert space of distributions 154
7.10 Bibliographical remarks . 164
8 Surface measures in Banach space 165
8.1 The measure derivatives and transition measures . 165
8.2 Surface measures 168
8.3 Algebraic properties of mapping \i \—► pp 173
8.4 Differentiation of surface measures . 176
8.5 Approximation of surface measures by volume measures 181
8.6 On integrability of second order surfaces in Banach space 188
8.7 Bibliographical remarks . . 191
9 Heat potentials on Hilbert space 193
9.1 Definition of heat potentials . 193
9.2 Differential properties of heat potentials . . . 200
9.3 Analyticity of heat potentials . . . . 203
9.4 Continuity of heat potentials . 208