About this Title
Ichiro Shigekawa, Kyoto University, Kyoto, Japan. Translated by Ichiro Shigekawa
Publication: Translations of Mathematical Monographs
Publication Year: 2004; Volume 224
ISBNs: 978-0-8218-2626-3 (print); 978-1-4704-4648-2 (online)
MathSciNet review: MR2060917
MSC: Primary 60-01; Secondary 46E30, 47D07, 60H05, 60H07, 60H10
Stochastic analysis is the analysis of functionals defined on the Wiener space, i.e., the space on which a Wiener process is realized. Since the Wiener space is infinite-dimensional, it requires a special calculus, the so-called Malliavin calculus.
The goal of the book is to provide the reader with a concise introduction to stochastic analysis, and, in particular, to the Malliavin calculus. The book contains a detailed description of all technical tools necessary to describe the theory, such as the Wiener process, the Ornstein–Uhlenbeck process, and Sobolev spaces. It also presents applications of stochastic calculus to the study of stochastic differential equations.
Graduate students and research mathematicians interested in probability and random processes.
Table of Contents
- Wiener space
- Orenstein-Uhlenbeck process
- The Littlewood-Paley-Stein inequality
- Sobolev spaces on an abstract Wiener space
- Absolute continuity of distributions and smoothness of density functions
- Application to stochastic differential equations
- Perspectives on current research