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Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes

With Emphasis on the Creation-Annihilation Techniques

Nicolas Bouleau, Laurent Denis (Autoren)

Buch | Hardcover

XVIII, 323 Seiten

2015 | 1st ed. 2015
Springer International Publishing (Verlag)
978-3-319-25818-8 (ISBN)

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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.

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.

Introduction.- Notations and Basic Analytical Properties.- 1.Reminders on Poisson Random Measures, Lévy Processes and Dirichlet Forms.- 2.Dirichlet Forms and (EID).- 3.Construction of the Dirichlet Structure on the Upper Space.- 4.The Lent Particle Formula and Related Formulae.- 5.Sobolev Spaces and Distributions on Poisson Space.- 6.- Space-Time Setting and Processes.- 7.Applications to Stochastic Differential Equations driven by a Random Measure.- 8.Affine Processes, Rates Models.- 9.Non Poissonian Cases.- A.Error Structures.- B.The Co-Area Formula.- References.

"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)

Erscheinungsdatum	29.12.2015
Reihe/Serie	Probability Theory and Stochastic Modelling
Zusatzinfo	XVIII, 323 p. 3 illus. in color.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	60H07, 60G57, 60G51, 60J45 • Dirichlet forms • lent particle method • Lévy processes • Malliavin calculus • mathematics and statistics • Probability theory and stochastic processes • random Poisson measures
ISBN-10	3-319-25818-4 / 3319258184
ISBN-13	978-3-319-25818-8 / 9783319258188
Zustand	Neuware