Probability Theory and Related Fields

, Volume 151, Issue 3–4, pp 403–433

Application of the lent particle method to Poisson-driven SDEs

Article

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bass R.F.: Uniqueness in law for pure jump Markov processes. Probab. Theory Relat. Fields 79, 271–287 (1988)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bichteler, K., Gravereaux, J.-B., Jacod, J.: Malliavin Calculus for Processes with Jumps (1987)Google Scholar
  3. 3.
    Bouleau N.: Error Calculus for Finance and Physics, the Language of Dirichlet Forms. De Gruyter, USA (2003)CrossRefGoogle Scholar
  4. 4.
    Bouleau N.: Error calculus and regularity of Poisson functionals: the lent particle method. C. R. Acad. Sc. Paris Mathématiques 346(13–14), 779–782 (2008)MathSciNetMATHGoogle Scholar
  5. 5.
    Bouleau N., Hirsch F.: Formes de Dirichlet générales et densité des variables aléatoires réelles sur l’espace de Wiener. J. Funct. Analysis 69(2), 229–259 (1986)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bouleau N., Hirsch F.: Dirichlet Forms and Analysis on Wiener Space. De Gruyter, USA (1991)MATHCrossRefGoogle Scholar
  7. 7.
    Bouleau N., Denis L.: Energy image density property and the lent particle method for Poisson measures. J. Funct. Analysis 257, 1144–1174 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bouleau N., Lépingle D.: Numerical Methods for Stochastic Processes. Wiley, New York (1994)MATHGoogle Scholar
  9. 9.
    Cancelier C., Chemin J.-Y.: Sous-ellipticité d’opérateurs intégro-différentiels vérifiant le principe du maximum Annali della Scuola Normale di Pisa. cl. d. Sc. ser4 20(2), 299–312 (1993)MathSciNetMATHGoogle Scholar
  10. 10.
    Carlen, E.A., Pardoux, E.: Differential calculus and integration by parts on Poisson space. In: Stochastics, Algebra and Analysis in Classical and Quantum Dynamics, pp. 63–73. Kluwer, Dordrecht (1990)Google Scholar
  11. 11.
    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
  12. 12.
    Denis L.: A criterion of density for solutions of Poisson-driven SDEs. Probab. Theory Relat. Fields 118, 406–426 (2000)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    El Karoui N., Lepeltier J.-P.: Représentation des processus ponctuels multivariés à l’aide d’un processus de Poisson. Z. Wahrsch. Verw. Geb. 39, 111–133 (1977)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Fournier N., Giet J.-S.: Existence of densities for jumping S.D.E.s. Stoch. Proc. Appl. 116(4), 643–661 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fukushima M., Oshima Y., Takeda M.: Dirichlet Forms and Symmetric Markov Processes. De Gruyter, USA (1994)MATHCrossRefGoogle Scholar
  16. 16.
    Hiraba S.: Existence and smoothness of transition density for jump-type Markov processes: application of Malliavin calculus. Kodai Math. j. 15, 28–49 (1992)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Ikeda N., Watanabe S.: Stochastic Differential Equation and Diffusion Processes. North-Holland, Koshanda (1981)Google Scholar
  18. 18.
    Ishikawa Y., Kunita H.: Malliavin calculus on the Wiener–Poisson space and its application to canonical SDE with jumps. Stoch. Process. Appl. 116, 1743–1769 (2006)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Jacob N.: Pseudo-Differential Operators and Markov Processes. Akademic Verlag, Berlin (1996)MATHGoogle Scholar
  20. 20.
    Jourdain B., Meleard S., Woyczynski W.: Nonlinear SDEs driven by Lévy processes and related PDEs. Alea 4, 1–29 (2008)MathSciNetMATHGoogle Scholar
  21. 21.
    Léandre R.: Régularité de processus de sauts dégénérés (I), (II). Ann. Inst. Henri Poincaré 21, 125–146 (1985)MATHGoogle Scholar
  22. 22.
    Léandre R.: Régularité de processus de sauts dégénérés (I), (II). Ann. Inst. Henri Poincaré 24, 209–236 (1988)MATHGoogle Scholar
  23. 23.
    Léandre, R.: Regularity of degenerated convolution semi-groups without use of the Poisson space (2007). Preprint at Inst. Mittag-LefflerGoogle Scholar
  24. 24.
    Malliavin P.: Sochastic Analysis. Springer, Berlin (1997)Google Scholar
  25. 25.
    Ma Z.M., Röckner M.: Construction of diffusion on configuration spaces. Osaka J. Math. 37, 273–314 (2000)MathSciNetMATHGoogle Scholar
  26. 26.
    Negoro A.: Stable-like processes: construction of the transition dentity and the behavior of the sample paths near t=0. Osaka J. Math. 31, 189–214 (1994)MathSciNetMATHGoogle Scholar
  27. 27.
    Nourdin I., Simon T.: On the absolute continuity of Lévy processes with drift. Ann. Probab. 34, 1035–1051 (2006)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Nualart, D., Vives, J.: Anticipative calculus for the Poisson process based on the Fock space. Sém. Prob. XXIV. Lect. Notes in M. vol. 1426. Springer, Berlin (1990)Google Scholar
  29. 29.
    Picard J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105, 481–511 (1996)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Privault N.: Equivalence of gradients on configuration spaces. Random Oper. Stoch. Equ. 7(3), 241–262 (1999)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Protter P.: Stochastic Integration and Differential Equations. Springer, Berlin (1990)MATHGoogle Scholar
  32. 32.
    Sato K.: Absolute continuity of multivariate distributions of class L. J. Multivar. Anal. 12, 89–94 (1982)MATHCrossRefGoogle Scholar
  33. 33.
    Sato K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  34. 34.
    Scotti, S.: Applications de la Théorie des Erreurs par Formes de Dirichlet, Thesis Univ. Paris-Est, Scuola Normale Pisa (2008). http://pastel.paristech.org/4501/
  35. 35.
    Song Sh.: Admissible vectors and their associated Dirichlet forms. Potential Analysis 1(4), 319–336 (1992)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Tsuchiya M.: Lévy measure with generalized polar decomposition and the associated SDE with jumps. Stoch. Stoch. Rep. 38, 95–117 (1992)MathSciNetMATHGoogle Scholar

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

Personalised recommendations