Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion

An Introduction

Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion

Assuming only basic knowledge of probability theory and functional analysis, this book provides a self-contained introduction to Malliavin calculus and infinite-dimensional Brownian motion. In an effort to demystify a subject thought to be difficult, it exploits the framework of nonstandard analysis, which allows infinite-dimensional problems to be treated as finite-dimensional. The result is an intuitive, indeed enjoyable, development of both Malliavin calculus and nonstandard analysis. The main aspects of stochastic analysis and Malliavin calculus are incorporated into this simplifying framework. Topics covered include Brownian motion, Ornstein–Uhlenbeck processes both with values in abstract Wiener spaces, Lévy processes, multiple stochastic integrals, chaos decomposition, Malliavin derivative, Clark–Ocone formula, Skorohod integral processes and Girsanov transformations. The careful exposition, which is neither too abstract nor too theoretical, makes this book accessible to graduate students, as well as to researchers interested in the techniques.


 Reviews:

"This book provides a self-contained exposition of Malliavin calculus for infinite-dimensional Brownian motion and for Lévy processes using nonstandard analysis techniques. This approach provides and alternative to the classical literature on the subject."
Anthony Réveillac, Mathematical Reviews

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