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3023 results in Probability theory and stochastic processes

Page 79 of 152

  • Preface

  • Ivan Nourdin, Université de Nancy I, France, Giovanni Peccati, Université du Luxembourg
  • Book:
    Normal Approximations with Malliavin Calculus
    Published online:
    05 June 2012
    Print publication:
    10 May 2012, pp xi-xiv

  • Summary

    This is a text about probabilistic approximations, which are mathematical statements providing estimates of the distance between the laws of two random objects. As the title suggests, we will be mainly interested in approximations involving one or more normal (equivalently called Gaussian) random elements. Normal approximations are naturally connected with central limit theorems (CLTs), i.e. convergence results displaying a Gaussian limit, and are one of the leading themes of the whole theory of probability.

    The main thread of our text concerns the normal approximations, as well as the corresponding CLTs, associated with random variables that are functionals of a given Gaussian field, such as a (fractional) Brownian motion on the real line. In particular, a pivotal role will be played by the elements of the socalled Gaussian Wiener chaos. The concept of Wiener chaos generalizes to an infinite-dimensional setting the properties of the Hermite polynomials (which are the orthogonal polynomials associated with the one-dimensional Gaussian distribution), and is now a crucial object in several branches of theoretical and applied Gaussian analysis.

    The cornerstone of our book is the combination of two probabilistic techniques, namely the Malliavin calculus of variations and Stein's method for probabilistic approximations.

    The Malliavin calculus of variations is an infinite-dimensional differential calculus, whose operators act on functionals of general Gaussian processes. Initiated by Paul Malliavin (starting from the seminal paper [69], which focused on a probabilistic proof of Hörmander's ‘sum of squares’ theorem), this theory is based on a powerful use of infinite-dimensional integration by parts formulae.

Trends in Stochastic Analysis
  • Edited by Jochen Blath, Peter Mörters, Michael Scheutzow
  • Published online:
    05 March 2012
    Print publication:
    09 April 2009
  • Presenting important trends in the field of stochastic analysis, this collection of thirteen articles provides an overview of recent developments and new results. Written by leading experts in the field, the articles cover a wide range of topics, ranging from an alternative set-up of rigorous probability to the sampling of conditioned diffusions. Applications in physics and biology are treated, with discussion of Feynman formulas, intermittency of Anderson models and genetic inference. A large number of the articles are topical surveys of probabilistic tools such as chaining techniques, and of research fields within stochastic analysis, including stochastic dynamics and multifractal analysis. Showcasing the diversity of research activities in the field, this book is essential reading for any student or researcher looking for a guide to modern trends in stochastic analysis and neighbouring fields.

Malliavin Calculus for Lévy Processes and Infinite-Dimensional Brownian Motion
  • Horst Osswald
  • Published online:
    05 March 2012
    Print publication:
    01 March 2012
  • 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.

Page 79 of 152