Probability Theory and Related Fields

, Volume 151, Issue 3–4, pp 403–433 | Cite as

Application of the lent particle method to Poisson-driven SDEs

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Abstract

We apply the Dirichlet form theory to stochastic differential equations with jumps as extension of Malliavin calculus reasoning. As in the continuous case, this weakens significantly the assumptions on the coefficients of the SDE. Thanks to the flexibility of the Dirichlet forms language, this approach brings also an important simplification which was neither available nor visible previously: an explicit formula giving the carré du champ matrix, i.e., the Malliavin matrix. Following this formula a new procedure appears, called the lent particle method which shortens the computations both theoretically and in concrete examples. In this paper which uses the construction done in Bouleau and Denis (J. Funct. Analysis 257:1144–1174, 2009) we restrict ourselves to the existence of densities; smoothness will be studied separately.

Keywords

Stochastic differential equation Poisson functional Dirichlet form Energy image density Lévy processes Gradient Carré du champ 

Mathematics Subject Classification (2000)

Primary 60G57 60H05 Secondary 60J45 60G51 
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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Ecole des PontsParis Tech, Paris-EstMarne-La-Vallée Cedex 2France
  2. 2.Equipe Analyse et ProbabilitésUniversité d’Evry-Val-d’EssonneEvry CedexFrance

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