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 OrnsteinUhlenbeck processes both with values in abstract Wiener spaces Lvy processes multiple stochastic integrals chaos decomposition Malliavin derivative ClarkOcone 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.