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Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion


Part of Cambridge Tracts in Mathematics

  • Date Published: March 2012
  • availability: Available
  • format: Hardback
  • isbn: 9781107016149

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About the Authors
  • 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.

    • Covers all the main aspects of stochastic analysis
    • A complex subject made accessible to graduate students
    • Very few prerequisites
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    Reviews & endorsements

    '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, 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 …' Zentralvlatt MATH

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    Product details

    • Date Published: March 2012
    • format: Hardback
    • isbn: 9781107016149
    • length: 428 pages
    • dimensions: 234 x 158 x 27 mm
    • weight: 0.76kg
    • availability: Available
  • Table of Contents

    Part I. The Fundamental Principles:
    1. Preface
    2. Martingales
    3. Fourier and Laplace transformations
    4. Abstract Wiener–Fréchet spaces
    5. Two concepts of no-anticipation in time
    6. Malliavin calculus on the space of real sequences
    7. Introduction to poly-saturated models of mathematics
    8. Extension of the real numbers and properties
    9. Topology
    10. Measure and integration on Loeb spaces
    Part II. An Introduction to Finite- and Infinite-Dimensional Stochastic Analysis:
    11. From finite- to infinite-dimensional Brownian motion
    12. The Itô integral for infinite-dimensional Brownian motion
    13. The iterated integral
    14. Infinite-dimensional Ornstein–Uhlenbeck processes
    15. Lindstrøm's construction of standard Lévy processes from discrete ones
    16. Stochastic integration for Lévy processes
    Part III. Malliavin Calculus:
    17. Chaos decomposition
    18. The Malliavin derivative
    19. The Skorokhod integral
    20. The interplay between derivative and integral
    21. Skorokhod integral processes
    22. Girsanov transformation
    23. Malliavin calculus for Lévy processes
    Appendix A. Poly-saturated models
    Appendix B. The existence of poly-saturated models

  • Author

    Horst Osswald, Universität Munchen
    Horst Osswald is a Professor of Mathematics at Universität München.

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