Probability Theory III: Stochastic Calculus

259 Pages·1998·6.225 MB·English

by  Yu. V. Prokhorov, A. N. Shiryaev (auth.)

Encyclopaedia of
Mathematical Sciences
Volume 45
Editor-in-Chief: R. V. Gamkrelidze
Springer-Verlag Berlin Heidelberg GmbH
Yu. V. Prokhorov A. N. Shiryaev
Probability Theory III
Stochastic Calculus
Springer
Consulting Editors of the Series:
A.A. Agrachev, A.A. Gonchar, E.F. Mishchenko,
N.M. Ostianu, V.P. Sakharova, A.B. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, VoI. 45,
Teoriya veroyatoastej - 3,
Publisher VINITI, Moscow 1989
Mathematics Subject Classification (1991):
60Hxx, 60G40, 60G44, 60G48, 60F17, 60J60, 6OJ65
ISSN 0938-0396
ISBN 978-3-642-08122-4 ISBN 978-3-662-03640-2 (eBook)
DOI 10.1007/978-3-662-03640-2
This work is subject to copyright. AII rights are reserved. whether the whole or part of the material is concemed.
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© Springer-Verlag Berlin Heidelberg 1998
Originally published by Springer-Verlag Berlin Heidelberg New York in 1998
Softcover reprint of the hardcover 1 st edition 1998
Typesetting: Camera·ready copy produced from the translators' input files
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SPIN 10020167 41/3143 -5432 I 0-Printed on acid-free paper
List of Editors, Authors and Translators
Editor-in-Chief
RV. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institu
te, u1. Gubkina 8, 117966 Moscow; Institute for Scientific Information
(VINITI), u1. Usievicha 20a, 125219 Moscow, Russia; e-mail:
[email protected]
Consulting Editors
Yu. V. Prokhorov, Steklov Mathematical Institute, u1. Gubkina 8, 117966
Moscow, Russia, e-mail: [email protected]
A. N. Shiryaev, Steklov Mathematical Institute, u1. Gubkina 8, 117966 Moscow,
Russia, e-mail: [email protected]
Authors
S. V. Anulova, Institut problem upravleniya RAN, u1. Provsoyuznaya 65,
117342 Moscow, Russia
N. V. Krylov, Department of Mathematics, University of Minnesota,
Minneapolis, MN 55455, USA, e-mail: [email protected]
R Sh. Liptser, Institute for Problems of Transmission of Information of the
Russian Academy of Sciences, u1. Ermolovoj 15, 101447 Moscow, Russia;
Department of Electrical Engineering-Systems, Tel Aviv University, Ramat
Aviv 69978, Israel, e-mail: [email protected]
A. N. Shiryaev, Steklov Mathematical Institute, u1. Gubkina 8, 117966 Moscow,
Russia, e-mail: [email protected]
A. Yu. Veretennikov, Inst. ofInformation Transmission Sciences, Russian
Academy of Sciences, 19 Bolshoy Karetnii, 101447 Moscow, Russia, e-mail:
[email protected], [email protected]
Translator
P. B. Slater, Community and Organization Research Institute, University of
California, Santa Barbara, CA 93106-2150, USA,
e-mail: [email protected]
Stochastic Calculus
S.V. Anulova, A.Yu. Veretennikov, N.V. Krylov,
R.Sh. Liptser, A.N. Shiryaev
Translated from the Russian
by P.B. Slater
Contents
Preface ....................................................... 5
Chapter 1. Introduction to Stochastic Calculus (N.V. Krylov) 7
§1. Brownian Motion and the Wiener Process ..................... 7
§2. Probabilistic Construction of the Solution of the Heat Equation.
Relation Between the Wiener Process and the Laplace Operator . 15
Ito
§3. The Integral and Differentiation Rules for Composite
Stochastic Functions ........................................ 19
§4. Stochastic Differential Equations and Diffusion Processes.
Girsanov's Theorem ........................................ 26
§5. Stochastic Differential Equations with Boundary Conditions ..... 32
References .................................................... 36
Chapter 2. Stochastic Differential and Evolution Equations .......... 38
I. Stochastic Differential Equations (SDEs) (S.V. Anulova,
A.Yu. Veretennikov) ............................................ 38
§1. Strong Solutions of Stochastic Differential Equations ........... 38
§2. Weak Solutions of Stochastic Differential Equations with
Non-Smooth Coefficients in Ed .............................. 50
§3. Differentiation of Solutions of SDEs with Respect to the Initial
Data ..................................................... 54
2 Stochastic Calculus
§4. Invariant Measure of a Diffusion Process ...................... 57
§5. Support of a Diffusion ...................................... 59
§6. Stochastic Differential Equations in Domains .................. 63
References ........................................... .1........ 72
II. Stochastic Evolution Equations (A.Yu. Veretennikov) ............ 76
§1. Introduction ............................................... 76
§2. Martingales and Stochastic Integrals in Hilbert Space ........... 77
§3. The Ito Formula for the Square of the Norm ................... 81
§4. Stochastic Differential Equations of Monotone Type in Banach
Spaces .................................................... 82
§5. Stochastic Partial Differential Equations I. The First
Boundary-Value Problem for Parabolic-Type Non-Linear
Equations ................................................. 85
§6. Stochastic Partial Differential Equations II. The Cauchy
Problem for Second-Order Linear Equations ................... 88
References .................................................... 90
III. Stochastic Calculus (Malliavin Calculus). Applications to
Stochastic Differential Equations (A.Yu. Veretennikov) . . . . . . . . . . . .. 91
§1. Introduction ............................................... 91
§2. Stochastic Derivatives ...................................... 92
§3. Rules of the Malliavin Calculus .............................. 96
§4. Smoothness of the Density (Scheme of the Proof) .............. 98
§5. The Bismut Approach 1 .................................... 99
§6. The Bismut Approach 2. Stochastic Differential Equations 101
§7. Stochastic Differential Equations (Smoothness of the Density
with Respect to Inverse Variables) . . . . . . . . . . . . . . . . . . . . . . . . . .. 107
References .................................................... 108
Chapter 3. Stochastic Calculus on Filtered Probability Spaces
(R.Sh. Liptser, A.N. Shiryaev) ................................... 111
I. Elements of the General Theory of Stochastic Processes ........... 111
§1. Kolmogorov's Axioms and the Stochastic Basis ................ 111
§2. Stopping Times, Adapted Stochastic Processes, Optional and
Predictable a-Algebras. Classification of Stopping Times ........ 112
§3. Martingales and Local Martingales ........................... 116
§4. Increasing Processes. Doob-Meyer Decomposition. Compensators . 118
§5. Random Measures. Integral Random Measures ................. 120
§6. Locally Square-Integrable Martingales. The Quadratic
Characteristic ............................................. 123
§7. Decomposition of Local Martingales .......................... 123
Contents 3
II. Semimartingales. Stochastic Integrals .......................... 125
§l. Semimartingales. Quadratic Variation. Quasimartingales ........ 125
§2. Construction of Stochastic Integrals with Respect to
Semimartingales ........................................... 127
§3. The Ito Formula ........................................... 129
§4. Construction of Stochastic Integrals with Respect to Random
Measures .................................................. 131
§5. Characteristics of Semimartingales. The Triple of Predictable
Characteristics T = (B, C, v). Martingale and Semimartingale
Problems. Examples. ....................................... 133
§6. Integral Representation of Local Martingales .................. 137
§7. Stability of the Class of Semimartingales with Respect to a
Series of Transformations ................................... 138
III. Absolute Continuity and Singularity of Probability Distributions . 140
§l. Local Density. Lebesgue Decomposition ....................... 140
§2. Girsanov's Theorem and its Generalization. Transformation of
Predictable Characteristics .................................. 141
§3. The Hellinger Integral and the Hellinger Process ............... 143
§4. General and Predictable Criteria of Absolute Continuity and
Singularity of Probability Measures ........................... 147
§5. Particular Cases ........................................... 148
Commentary to Chapter 3 ...................................... 152
References .................................................... 154
Chapter 4. Martingales and Limit Theorems for Stochastic Processes
(R.Sh. Liptser, A.N. Shiryaev) ................................... 158
I. Theory: Weak Convergence of Probability Measures
on Metric Spaces ............................................... 158
§l. Introduction ............................................... 158
§2. Different Types of Convergence. Skorokhod Topology ........... 160
§3. Brief Review of a Number of Classical Limit Theorems of
Probability Theory ......................................... 166
§4. Convergence of Processes with Independent Increments ......... 178
§5. The Convergence of Semimartingales to Processes with
Independent Increments ..................................... 188
§6. Relative Compactness and Tightness of Families of Distributions
of Semimartingales ......................................... 200
§7. Convergence of Semimartingales to a Semimartingale ........... 202
§8. The Martingale Problem .................................... 210
4 Stochastic Calculus
II. Applications: The Invariance Principle
and Diffusion Approximation .................................... 213
§l. The Invariance Principle for Stationary and Markov Processes ... 213
§2. The Stochastic Averaging Principle in Models without Diffusion 226
§3. Diffusion Approximation of Semimartingales. The Averaging
Principle in Models with Diffusion ............................ 229
§4. Diffusion Approximation for Systems with Physical White Noise 233
§5. Diffusion Approximation for Semimartingales with Normal
Reflection in a Convex Domain .............................. 237
Commentary to Chapter 4 ...................................... 243
References .................................................... 244
Author Index 249
Subject Index 251
Preface 5
Preface
In the axioms of probability theory proposed by Kolmogorov the basic
"probabilistic" object is the concept of a probability model or probability
space. This is a triple (n, F, P), where n is the space of elementary events
n
or outcomes, F is a a-algebra of subsets of announced by the events and
P is a probability measure or a probability on the measure space (n, F).
This generally accepted system of axioms of probability theory proved to be
so successful that, apart from its simplicity, it enabled one to embrace the
classical branches of probability theory and, at the same time, it paved the
way for the development of new chapters in it, in particular, the theory of
random (or stochastic) processes.
In the theory of random processes, various classes of processes have been
studied in depth. Theories of processes with independent increments, Markov
processes, stationary processes, among others, have been constructed. In the
formation and development of the theory of random processes, a significant
event was the realization that the construction of a "general theory of ran
dom processes" requires the introduction of a flow of a-algebras (a filtration)
F = (Ftk::o supplementing the triple (n, F, P), where Ft is interpreted as
the collection of events from F observable up to time t.
It is this assumption of the presence on (n, F, P) of a flow F = (Ftk~o
that has given rise to such objects as Markov times or stopping times, adapted
processes, optional and predictable a-algebras, martingales, local martingales,
semimartingales, the stochastic integral, the Ito change of variables formula,
etc., which are the ingredients of the theory of stochastic calculus.
Thus, stochastic calculus axiomatizes the concept of a stochastic basis
B = (n, F, F = (Ftk::o, P),
which lies at the basis of our entire discussion. Here (n, F, P) is a probability
space, and F is some distinguished flow of a-algebras.
The formation of the concept of a "stochastic basis" as one of the specifica
tions of a probability space passed through many stages of discussion of par
ticular cases, refinements, generalizations, etc. Here the theory of stochastic
integration for Brownian motion and a centered Poisson measure, developed
by Ito, was crucial.
In the first chapter of this volume, an introduction is given to stochastic cal
culus which is called upon for presenting various aspects of Brownian motion
and its connection with the theory of partial differential equations, the latter
being the fundamental principle of Kolmogorov's classical paper "Analytical
Methods in Probability Theory" .
Stochastic integration theory has been brought to perfection for the most
part in random processes that are solutions of stochastic differential equa
tions which, as we have said, are a particular case of semimartingales - that