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  3. Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion
  • Cited by 8
Publisher:
Cambridge University Press
Online publication date:
March 2012
Print publication year:
2012
Online ISBN:
9781139060110
DOI:
https://doi.org/10.1017/CBO9781139060110
Subjects:
Mathematics (general), Mathematics, Logic, Categories and Sets, Probability Theory and Stochastic Processes, Statistics and Probability
Series:
Cambridge Tracts in Mathematics (191)
Information

Book description

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 [an] alternative to the classical literature on the subject.'

Anthony Réveillac Source: Mathematical Reviews

'In addition to being self-contained, this book remains at an accessible level despite the amount of material to be assimilated on the *-extension of real numbers. It covers the main aspects of the Malliavin calculus and succeeds in providing a good global understanding …'

Source: Zentralvlatt MATH

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