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© 2015

Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes

With Emphasis on the Creation-Annihilation Techniques

Book

Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 76)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Nicolas Bouleau, Laurent Denis
    Pages 1-8
  3. Nicolas Bouleau, Laurent Denis
    Pages 9-29
  4. Nicolas Bouleau, Laurent Denis
    Pages 31-39
  5. Nicolas Bouleau, Laurent Denis
    Pages 41-81
  6. Nicolas Bouleau, Laurent Denis
    Pages 83-105
  7. Nicolas Bouleau, Laurent Denis
    Pages 107-135
  8. Nicolas Bouleau, Laurent Denis
    Pages 137-170
  9. Nicolas Bouleau, Laurent Denis
    Pages 229-238
  10. Nicolas Bouleau, Laurent Denis
    Pages 239-264
  11. Back Matter
    Pages 265-323

About this book

Introduction

A simplified approach to Malliavin calculus adapted to Poisson random measures is developed and applied in this book. Called the “lent particle method” it is based on perturbation of the position of particles. Poisson random measures describe phenomena involving random jumps (for instance in mathematical finance) or the random distribution of particles (as in statistical physics). Thanks to the theory of Dirichlet forms, the authors develop a mathematical tool for a quite general class of random Poisson measures and significantly simplify computations of Malliavin matrices of Poisson functionals. The method gives rise to a new explicit calculus that they illustrate on various examples: it consists in adding a particle and then removing it after computing the gradient. Using this method, one can establish absolute continuity of Poisson functionals such as Lévy areas, solutions of SDEs driven by Poisson measure and, by iteration, obtain regularity of laws. The authors also give applications to error calculus theory. This book will be of interest to researchers and graduate students in the fields of stochastic analysis and finance, and in the domain of statistical physics. Professors preparing courses on these topics will also find it useful. The prerequisite is a knowledge of probability theory.

Keywords

60H07, 60G57, 60G51, 60J45 Dirichlet forms Lévy processes Malliavin calculus lent particle method random Poisson measures

Authors and affiliations

  1. 1.Université Paris-Est L'École des Ponts ParisTechMarne la ValléeFrance
  2. 2.Laboratoire Manceau de MathématiquesUniversité du MaineLe MansFrance

About the authors

Laurent Denis is currently professor at the Université du Maine. He has been head of the department of mathematics at the University of Evry (France). He is a specialist in Malliavin calculus, the theory of stochastic partial differential equations and mathematical finance.

Nicolas Bouleau is emeritus professor at the Ecole des Ponts ParisTech. He is known for his works in potential theory and on Dirichlet forms with which he transformed the approach to error calculus. He has written more than a hundred articles and several books on mathematics and on other subjects related to the philosophy of science. He holds several awards including the Montyon prize from the French Academy of Sciences and is a member of the Scientific Council of the Nicolas Hulot Foundation.

Bibliographic information

  • Book Title Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes
  • Book Subtitle With Emphasis on the Creation-Annihilation Techniques
  • Authors Nicolas Bouleau
    Laurent Denis
  • Series Title Probability Theory and Stochastic Modelling
  • Series Abbreviated Title Probability and Stochastic (formerly: PIA & SMAP)
  • DOI https://doi.org/10.1007/978-3-319-25820-1
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-25818-8
  • Softcover ISBN 978-3-319-79845-5
  • eBook ISBN 978-3-319-25820-1
  • Series ISSN 2199-3130
  • Series E-ISSN 2199-3149
  • Edition Number 1
  • Number of Pages XVIII, 323
  • Number of Illustrations 0 b/w illustrations, 3 illustrations in colour
  • Topics Probability Theory and Stochastic Processes
  • Buy this book on publisher's site
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Reviews

“This book is based on a course given at the Institute Henri Poincare in Paris, in 2011. … this is a deep book that is very well written and could be interesting to anybody working with jump diffusion stochastic models.” (Josep Vives, Mathematical Reviews, February, 2017)