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The Malliavin Calculus and Related Topics

  • Book
  • © 2006

Overview

Authors:
  1. David Nualart

    1. Department of Mathematics, University of Kansas, Lawrence, U.S.A.

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  • Can be used as a basis of a lecture course on Malliavin Calculus, because the exposition, being self-contained, is quite accessible
  • Contains a great number of exercises
  • A survey of the applications of Malliavin Calculus to the stochastic calculus with respect to fractional Brownian motion and to financial mathematics makes this book very useful to anyone interested in the current research in the field
  • Includes supplementary material: sn.pub/extras

Part of the book series: Probability and Its Applications (PIA)

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Table of contents (8 chapters)

  1. Front Matter

    Pages I-XIV
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  2. Introduction

    • David Nualart
    Pages 1-2

  3. Analysis on the Wiener space

    • David Nualart
    Pages 3-83

  4. Regularity of probability laws

    • David Nualart
    Pages 85-167

  5. Anticipating stochastic calculus

    • David Nualart
    Pages 169-223

  6. Transformations of the Wiener measure

    • David Nualart
    Pages 225-271

  7. Fractional Brownian motion

    • David Nualart
    Pages 273-320

  8. Malliavin Calculus in finance

    • David Nualart
    Pages 321-349

  9. Malliavin Calculus in finance

    Pages 321-349

  10. Back Matter

    Pages 351-382
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Keywords

About this book

There have been ten years since the publication of the ?rst edition of this book. Since then, new applications and developments of the Malliavin c- culus have appeared. In preparing this second edition we have taken into account some of these new applications, and in this spirit, the book has two additional chapters that deal with the following two topics: Fractional Brownian motion and Mathematical Finance. The presentation of the Malliavin calculus has been slightly modi?ed at some points, where we have taken advantage of the material from the lecturesgiveninSaintFlourin1995(seereference[248]).Themainchanges and additional material are the following: In Chapter 1, the derivative and divergence operators are introduced in the framework of an isonormal Gaussian process associated with a general 2 Hilbert space H. The case where H is an L -space is trated in detail aft- s,p wards (white noise case). The Sobolev spaces D , with s is an arbitrary real number, are introduced following Watanabe’s work. Chapter2includesageneralestimateforthedensityofaone-dimensional random variable, with application to stochastic integrals. Also, the c- position of tempered distributions with nondegenerate random vectors is discussed following Watanabe’s ideas. This provides an alternative proof of the smoothness of densities for nondegenerate random vectors. Some properties of the support of the law are also presented.

Reviews

From the reviews of the second edition:

"Nualart’s book serves both pedagogic and research needs. On the one hand, it is written to teach the subject. … On the other hand, the applications in the book are sufficiently broad and in depth that the reader who masters them should be prepared for research. … Furthermore, the unified approach and the careful statement of technical results in the development of the applications make the text a handy reference for researchers. … The bibliography is extensive and has been updated." (Daniel Ocone, Mathematical Reviews, Issue 2006 j)

Authors and Affiliations

  • Department of Mathematics, University of Kansas, Lawrence, U.S.A.

    David Nualart

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