Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes

1. Front Matter

   Pages i-xviii

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2. Introduction

   • Nicolas Bouleau, Laurent Denis

   Pages 1-8

3. Introduction to the Theory of Dirichlet Forms

   • Nicolas Bouleau, Laurent Denis

   Pages 9-29

4. Reminders on Poisson Random Measures

   • Nicolas Bouleau, Laurent Denis

   Pages 31-39

5. Construction of the Dirichlet Structure on the Upper Space

   • Nicolas Bouleau, Laurent Denis

   Pages 41-81

6. The Lent Particle Formula

   • Nicolas Bouleau, Laurent Denis

   Pages 83-105

7. Space-Time Setting and Examples

   • Nicolas Bouleau, Laurent Denis

   Pages 107-135

8. Sobolev Spaces and Distributions on Poisson Space

   • Nicolas Bouleau, Laurent Denis

   Pages 137-170

9. Applications to Stochastic Differential Equations Driven by a Random Measure

   • Nicolas Bouleau, Laurent Denis

   Pages 171-228

10. Affine Processes, Rates Models

    • Nicolas Bouleau, Laurent Denis

    Pages 229-238

11. Non Poissonian Cases

    • Nicolas Bouleau, Laurent Denis

    Pages 239-264

12. Back Matter

    Pages 265-323

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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.

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

“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)

• Université Paris-Est L'École des Ponts ParisTech, Marne la Vallée, France

  Nicolas Bouleau

• Laboratoire Manceau de Mathématiques, Université du Maine, Le Mans, France

  Laurent Denis

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.

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