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Malliavin calculus for Lévy processes and infinite-dimensional Brownian motion : an introduction / Horst Osswald.


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Osswald, Horst.
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Cambridge ; New York : Cambridge University Press, 2012, ©2012.
  • Book
  • xix, 407 pages : illustrations ; 24 cm.
Includes bibliographical references (pages 398-403) and index.
  • Martingales
  • Fourier and Laplace transformations
  • Abstract Wiener-Fréchet spaces
  • Two concepts of no-anticipation in time
  • Malliavin calculus on the space of real sequences
  • Introduction to poly-saturated models of mathematics
  • Extension of the real numbers
  • Topology
  • Measure and integration on Loeb spaces
  • From finite- to infinite-dimensional Brownian motion
  • The Itô integral for infinite-dimensional Brownian motion
  • Multiple integrals
  • Infinite-dimensional Ornstein-Uhlenbeck processes
  • Lindstrøm's construction of standard Lévy processes from discrete ones
  • Stochastic integration for Lévy processes
  • Chaos decomposition (for infinite-dimensional Brownian motion)
  • The Malliavin derivative
  • The Skorohod integral
  • The interplay between derivative and integral
  • Skorohod integral processes
  • Girsanov transformations
  • Malliavin calculus for Lévy processes (Chaos, Malliavin derivative, Clark-Ocone formula, Skorohod integral processes, Smooth representations, a communication rule for derivative and limit, product-, chainrule, Girsanov transformations)
  • Poly-saturated models
  • The existence of poly-saturated models.
"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"-- Provided by publisher.
Cambridge tracts in mathematics ; 191.

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