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Probability Theory and Related Fields

, Volume 105, Issue 4, pp 481–511 | Cite as

On the existence of smooth densities for jump processes

  • Jean Picard
Article

Summary

We consider a Lévy process X t and the solution Y t of a stochastic differential equation driven by X t; we suppose that X t has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the existence of a smooth density for Y t: these conditions are similar to those of the classical Malliavin calculus for continuous diffusions. More generally, we study the smoothness of the law of variables F defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function of F.

Mathematics Subject Classification (1991)

60H07 60J75 60J30 

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Jean Picard
    • 1
  1. 1.Laboratoire de Mathématiques Appliquées, URA 1501 du CNRSUniversité Blaise PascalAubière CedexFrance

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