Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
On the existence of smooth densities for jump processes
Download PDF
Download PDF
  • Published: December 1996

On the existence of smooth densities for jump processes

  • Jean Picard1 

Probability Theory and Related Fields volume 105, pages 481–511 (1996)Cite this article

  • 394 Accesses

  • 116 Citations

  • Metrics details

An Erratum to this article was published on 18 February 2010

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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bichteler, K., Gravereaux, J.B., Jacob, J.: Malliavin calculus for processes with jumps. Stochastics Monographs, vol. 2, London: Gordon and Breach 1987

    MATH  Google Scholar 

  2. Bismut, J.M.: Calcul des variations stochastique et processus de sauts. Z. Wahrscheinlichkeitstheorie Verw. Geb.63, 147–235 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  3. Carlen, E., Pardoux, E.: Differential calculus and integration by parts on Poisson space. In: Stochastics, Algebra and Analysis in Classical and Quantum Dynamics (Marseille, 1988), pp. 63–73 (Math. Appl., Vol 59) Dordrecht: Kluwer 1990

    Google Scholar 

  4. Fujiwara, T., Kunita, H.: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ.25, 71–106 (1985)

    MATH  MathSciNet  Google Scholar 

  5. Léandre, R.: Flot d'une équation différentielle stochastique avec semi-martingale directrice discontinue. In: Séminaire de Probabilités XIX, pp. 271–274 (Lect. Notes Math., Vol. 1123) Berlin: Springer 1985

    Google Scholar 

  6. Léandre, R.: Régularité de processus de sauts dégénérés. Ann. Inst. Henri Poincaré Prob. Stat.21, 125–146 (1985).

    MATH  Google Scholar 

  7. Léandre, R.: Régularité de processus de sauts dégénérés (II). Ann. Inst. Henri Poincaré Prob. Stat.24, 209–236 (1988)

    MATH  Google Scholar 

  8. Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators. In: Proc. Intern. Symp. SDE (Kyoto, 1976), pp. 195–263. New York: Wiley 1978

    Google Scholar 

  9. Meyer, P.A.: Flot d'une équation différentielle stochastique. In: Séminaire de Probabilités XV, pp. 103–117, (Lect. Notes Math., Vol. 850) Berlin: Springer 1981

    Google Scholar 

  10. Nualart, D., Vives, J.: Anticipative calculus for the Poisson process based on the Fock space, In: Séminaire de Probabilités XXIV, pp. 154–165 (Lect. Notes Math., Vol. 1426) Berlin: Springer 1990

    Google Scholar 

  11. Nualart, D., Vives, J.: A duality formula on the Poisson space and some applications. In: Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), (Prog. Probab., Vol. 36) Basel: Birkhäuser 1995

    Google Scholar 

  12. Picard, J.: Formules de dualité sur l'espace de Poisson. Ann. Inst. Henri Poincaré Prob. Stat., to appear

  13. Sato, K.: Absolute continuity of multivariate distributions of class L. J. Multivariate Anal.12, 89–94 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Tucker, H.G.: On a necessary and sufficient condition that an infinitely divisible distribution be absolutely continuous. Trans. Amer. Math. Soc.118, 316–330 (1965)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées, URA 1501 du CNRS, Université Blaise Pascal, F-63177, Aubière Cedex, France

    Jean Picard

Authors
  1. Jean Picard
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s00440-010-0267-x.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Picard, J. On the existence of smooth densities for jump processes. Probab. Th. Rel. Fields 105, 481–511 (1996). https://doi.org/10.1007/BF01191910

Download citation

  • Received: 07 December 1994

  • Revised: 22 January 1996

  • Issue Date: December 1996

  • DOI: https://doi.org/10.1007/BF01191910

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (1991)

  • 60H07
  • 60J75
  • 60J30
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature