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.
References
Bass R.F.: Uniqueness in law for pure jump Markov processes. Probab. Theory Relat. Fields 79, 271–287 (1988)
Bichteler, K., Gravereaux, J.-B., Jacod, J.: Malliavin Calculus for Processes with Jumps (1987)
Bouleau N.: Error Calculus for Finance and Physics, the Language of Dirichlet Forms. De Gruyter, USA (2003)
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)
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)
Bouleau N., Hirsch F.: Dirichlet Forms and Analysis on Wiener Space. De Gruyter, USA (1991)
Bouleau N., Denis L.: Energy image density property and the lent particle method for Poisson measures. J. Funct. Analysis 257, 1144–1174 (2009)
Bouleau N., Lépingle D.: Numerical Methods for Stochastic Processes. Wiley, New York (1994)
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)
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)
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)
Denis L.: A criterion of density for solutions of Poisson-driven SDEs. Probab. Theory Relat. Fields 118, 406–426 (2000)
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)
Fournier N., Giet J.-S.: Existence of densities for jumping S.D.E.s. Stoch. Proc. Appl. 116(4), 643–661 (2005)
Fukushima M., Oshima Y., Takeda M.: Dirichlet Forms and Symmetric Markov Processes. De Gruyter, USA (1994)
Hiraba S.: Existence and smoothness of transition density for jump-type Markov processes: application of Malliavin calculus. Kodai Math. j. 15, 28–49 (1992)
Ikeda N., Watanabe S.: Stochastic Differential Equation and Diffusion Processes. North-Holland, Koshanda (1981)
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)
Jacob N.: Pseudo-Differential Operators and Markov Processes. Akademic Verlag, Berlin (1996)
Jourdain B., Meleard S., Woyczynski W.: Nonlinear SDEs driven by Lévy processes and related PDEs. Alea 4, 1–29 (2008)
Léandre R.: Régularité de processus de sauts dégénérés (I), (II). Ann. Inst. Henri Poincaré 21, 125–146 (1985)
Léandre R.: Régularité de processus de sauts dégénérés (I), (II). Ann. Inst. Henri Poincaré 24, 209–236 (1988)
Léandre, R.: Regularity of degenerated convolution semi-groups without use of the Poisson space (2007). Preprint at Inst. Mittag-Leffler
Malliavin P.: Sochastic Analysis. Springer, Berlin (1997)
Ma Z.M., Röckner M.: Construction of diffusion on configuration spaces. Osaka J. Math. 37, 273–314 (2000)
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)
Nourdin I., Simon T.: On the absolute continuity of Lévy processes with drift. Ann. Probab. 34, 1035–1051 (2006)
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)
Picard J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105, 481–511 (1996)
Privault N.: Equivalence of gradients on configuration spaces. Random Oper. Stoch. Equ. 7(3), 241–262 (1999)
Protter P.: Stochastic Integration and Differential Equations. Springer, Berlin (1990)
Sato K.: Absolute continuity of multivariate distributions of class L. J. Multivar. Anal. 12, 89–94 (1982)
Sato K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
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/
Song Sh.: Admissible vectors and their associated Dirichlet forms. Potential Analysis 1(4), 319–336 (1992)
Tsuchiya M.: Lévy measure with generalized polar decomposition and the associated SDE with jumps. Stoch. Stoch. Rep. 38, 95–117 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0303-x
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