Diffusion processes serve as a mathematical model for the physical phenomenon of diffusion. One of the most important problems in the theory of diffusion processes is the development of methods for constructing these processes from a given diffusion matrix and a given drift vector. Focusing on the investigation of this problem, this book is intended for specialists in the theory of random processes and its applications.
A generalized diffusion process (that is, a continuous Markov process for which the Kolmogorov local characteristics exist in the generalized sense) can serve as a model for diffusion in a medium moving in a nonregular way. The author constructs generalized diffusion processes under two assumptions: first, that the diffusion matrix is sufficiently regular; and second, that the drift vector is a function integrable to some power, or is a generalized function of the type of the derivative of a measure.
Author(s): N. I. Portenko
Series: Translations of Mathematical Monographs, Vol. 83
Publisher: American Mathematical Society
Year: 1990
Language: English
Pages: C+x+180+B
Cover
Generalized Diffusion Processes
Copyright ©1990 by the American Mathematical Society
ISBN 0-8218-4538-1
QA274.75.P6713 1990 519.2'33-dc20
LCCN 90-21198 CIP
Contents
Preface to the English Edition
Preface
CHAPTER I The Method of Absolutely Continuous Change of Measure
§1. A lemma on the existence of exponential moments
§2. An inequality for solutions of stochastic differential equations with zero drift vector
§3. Properties of exponential supermartingales
§4. Construction of a solution in the case of Conditions (A) and (B)
§5. Properties of the solution constructed. Uniqueness
§6. A limit theorem
§7. Construction of a solution in the general case
CHAPTER II The Analytic Method
§1. Two lemmas
§2. The fundamental solution
§3. Construction of a solution of a stochastic differential equation
§4. A limit theorem
§5. The homogeneous case
§6. The case m = 1 and b (x) - 1
CHAPTER III Generalized Diffusion Processes
§ 1. Definitions
§2. Processes with integrable drift coefficient
§3. Processes with generalized drift coefficient
§4. Stochastic differential equations with generalized drift vector
Comments
Chapter 1
Chapter 2
Chapter 3
Bibliography
Subject Index
Back Cover
