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The Lent Particle Method for Marked Point Processes

  • Nicolas Bouleau
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2006)

Abstract

Although introduced in the case of Poisson random measures (cf. Bouleau and Denis [2, 3]), the lent particle method applies as well in other situations. We study here the case of marked point processes. In this case the Malliavin calculus (here in the sense of Dirichlet forms) operates on the marks and the point process does not need to be Poisson. The proof of the method is even much simpler than in the case of Poisson random measures. We give applications to isotropic processes and to processes whose jumps are modified by independent diffusions.

Poisson random measure Lent particle method Marked point process Isotropic process Dirichlet form Energy Image Density property Lvy process Wiener space Ornstein-Uhlenbeck form Malliavin calculus 

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References

  1. 1.
    Bouleau, N.: Décomposition de l’énergie par niveau de potentiel. In: Lecture Notes in Mathematics, vol. 1096, Springer, Berlin (1984)Google Scholar
  2. 2.
    Bouleau, N., Denis, L.: Energy image density property and local gradient for Poisson random measures. J. Funct. Anal. 257(4), 1144–1174 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bouleau, N., Denis, L.: Application of the lent particle method to Poisson driven SDE’s. Probab. Theor. Relat. Field (2010) online first DOI 10.1007/s00440-010-0303-xGoogle Scholar
  4. 4.
    Bouleau, N., Hirsch, F.: Formes de Dirichlet générales et densité des variables aléatoires réelles sur l’espace de Wiener. J. Funct. Anal. 69(2), 229–259 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. De Gruyter, Berlin (1991)zbMATHCrossRefGoogle Scholar
  6. 6.
    Coquio, A.: Formes de Dirichlet sur l’espace canonique de Poisson et application aux équations différentielles stochastiques. Ann. Inst. Henri. Poincaré 19(1), 1–36 (1993)MathSciNetGoogle Scholar
  7. 7.
    Sato, K.-I.: Absolute continuity of multivariate distributions of class L. J. Multivariate Anal. 12(1), 89–94 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Song, S.: Admissible vectors and their associated Dirichlet forms. Potential Anal. 1(4), 319–336 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Ecole des Ponts ParisTechParisFrance

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