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Selected Topics in Malliavin Calculus

Chaos, Divergence and So Much More

  • Textbook
  • © 2022

Overview

Authors:
  1. Laurent Decreusefond

    1. Telecom Paris, Institut Polytechnique de Paris, Paris, France

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  • Provides a pedagogical and self-contained introduction to Malliavin calculus
  • Covers some of the most modern applications of Malliavin calculus like the Malliavin-Stein method
  • Illustrates the links between Malliavin calculus and functional analysis in measure space

Part of the book series: Bocconi & Springer Series (BS, volume 10)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Wiener Space

    • Laurent Decreusefond
    Pages 1-24

  3. Gradient and Divergence

    • Laurent Decreusefond
    Pages 25-59

  4. Wiener Chaos

    • Laurent Decreusefond
    Pages 61-88

  5. Fractional Brownian Motion

    • Laurent Decreusefond
    Pages 89-120

  6. Poisson Space

    • Laurent Decreusefond
    Pages 121-148

  7. The Malliavin–Stein Method

    • Laurent Decreusefond
    Pages 149-169

  8. Back Matter

    Pages 171-172
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Keywords

About this book

This book is not a research monograph about Malliavin calculus with the latest results and the most sophisticated proofs. It does not contain all the results which are known even for the basic subjects which are addressed here. The goal was to give the largest possible variety of proof techniques. For instance, we did not focus on the proof of concentration inequality for functionals of the Brownian motion, as it closely follows the lines of the analog result for Poisson functionals. This book grew from the graduate courses I gave at Paris-Sorbonne and Paris-Saclay universities, during the last few years. It is supposed to be as accessible as possible for students who have  knowledge of Itô calculus and some rudiments of functional analysis.

Reviews

“For all the notions, precise definitions of the mathematical objects and complete proofs of the results are given; some examples are thoroughly studied. At the end of each chapter, the reader can also find some classical mathematical background. This book should prove useful for students and academic scholars wishing to get acquainted with Malliavin’s calculus, and also for more advanced researchers interested by more recent applications.” (Jean Picard, zbMATH 1514.60002, 2023)

Authors and Affiliations

  • Telecom Paris, Institut Polytechnique de Paris, Paris, France

    Laurent Decreusefond

About the author

Laurent Decreusefond is a former student of Ecole Normale Supérieure de Paris-Saclay. He received the Agrégation in 1989, his PhD in 1994 and his Habilitation in 2001 in Mathematics. He is now a full professor of Mathematics at the Institut Polytechnique de Paris, one of the most renowned French research and teaching institutions.


His research topics are twofold. The theoretical part is devoted to Malliavin calculus and its applications. He is the author of a highly cited paper about fractional Brownian motion that paved the way to a thousand research articles.   Recently, he has been interested in the functional Stein-Malliavin method, which gives the convergence rate in functional limit theorems.


On a more applied part, he proposed new paradigms for stochastic modelling of telecom systems, including stochastic geometry and random topological algebra.


He coauthored several papers that gavea new approach to the coverage analysis of cellular systems. The performance of some of the algorithms so defined may be analysed with mathematical tools coming from the Malliavin calculus, such as concentration inequalities.

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