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Application of the lent particle method to Poisson-driven SDEs
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  • Published: 23 June 2010

Application of the lent particle method to Poisson-driven SDEs

  • Nicolas Bouleau1 &
  • Laurent Denis2 

Probability Theory and Related Fields volume 151, pages 403–433 (2011)Cite this article

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  • 9 Citations

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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.

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Author information

Authors and Affiliations

  1. Ecole des Ponts, Paris Tech, Paris-Est, 6 Avenue Blaise Pascal, 77455, Marne-La-Vallée Cedex 2, France

    Nicolas Bouleau

  2. Equipe Analyse et Probabilités, Université d’Evry-Val-d’Essonne, Boulevard François Mitterrand, 91025, Evry Cedex, France

    Laurent Denis

Authors
  1. Nicolas Bouleau
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  2. Laurent Denis
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Correspondence to Laurent Denis.

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Bouleau, N., Denis, L. Application of the lent particle method to Poisson-driven SDEs. Probab. Theory Relat. Fields 151, 403–433 (2011). https://doi.org/10.1007/s00440-010-0303-x

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  • Received: 13 April 2009

  • Revised: 10 February 2010

  • Published: 23 June 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00440-010-0303-x

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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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