Stochastic analysis is often understood as the analysis of functionals defined on the Wiener space, i.e., the space on which the Wiener process is realized. Since the Wiener space is infinite-dimensional, it requires a special calculus, the so-called Malliavin calculus.

This book provides readers with a concise introduction to stochastic analysis, in particular, to the Malliavin calculus. It contains a detailed description of all the 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. The volume is suitable for graduate students and research mathematicians interested in probability and random processes.

Readership

Graduate students and research mathematicians interested in probability and random processes.

Reviews

"The book is wonderfully organized ... clearly written ... should be a useful text for an informal seminar or course on these developments. ... Shigekawa is to be congratulated on writing a nice book and the AMS for bringing it out at a reasonable price."

-- Bulletin of the American Mathematical Society

Table of Contents

• Wiener space
• Orenstein-Uhlenbeck process
• The Littlewood-Paley-Stein inequality
• Sobolev spaces on an abstrct Wiener space
• Absolute continuity of distributions and smoothness of density functions
• Application to stochastic differential equations
• Perspectives on current research
• Bibliography
• Index