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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 1023–1050 | Cite as

Regularity of the Law of Stochastic Differential Equations with Jumps Under Hörmander’s Conditions: The Lent Particle Method

  • Jiagang Ren
  • Hua ZhangEmail author
Article
  • 107 Downloads

Abstract

In this paper, we study the regularity of the laws of stochastic differential equations with jumps using the recently well-developed lent particle method introduced by Bouleau and Denis under some kind of Hörmander’s conditions.

Keywords

Regularity Density Hörmander’s condition Stochastic differential equations Lévy processes Poisson functional Dirichlet form Gradient Carré du champ 

Mathematics Subject Classification (2010)

60H10 60G51 

References

  1. 1.
    Bouleau, N., Denis, L.: Energy image density property and the lent particle method for Poisson measure. J. Funct. Anal. 257, 1144–1174 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bouleau, N., Denis, L.: Application of the lent particle method to Poisson-driven SDEs. Probab. Theory Relat. Fields 151, 403–433 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bouleau, N., Denis, L.: Iteration of the lent particle method for existence of smooth densities of Poisson functionals. Potential Anal. 38, 169–205 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bouleau, N., Denis, L.: Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bichteler, K., Gravereaux, J.B., Jacod, J.: Malliavin Calculus for Processes with Jumps. Gordan and Breach Science Publishers, London (1987)zbMATHGoogle Scholar
  6. 6.
    Carlen, E.A., Pardoux, E.: Differential calculus and integration by parts on Poisson space. In: Albeverio, S., Blanchard, P., Testard, D. (eds.) Stochastic, Algebra and Analysis in Classical and Quantum Dynamics, pp. 63–73. Kluwer, Dordrecht (1990)Google Scholar
  7. 7.
    Dellacherié, D., Meyer, P.A.: Probability and Potential, North-Holland Mathematics Studies, vol. 29. North-Holland Publishing Co., Amsterdam (1978)Google Scholar
  8. 8.
    Léandre, R.: Régularité de processus de sauts dégénérés (II). Ann. Inst. Henri Poincaré 24, 209–236 (1988)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. In: Proceedings of the International Conference on Stochastic Differential Equations, Kyoto, pp. 195–263. Kinokuniya (1978)Google Scholar
  10. 10.
    Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  11. 11.
    Picard, J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105, 481–511 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  13. 13.
    Shigekawa, I.: Stochastic Analysis. American Mathematical Society, Providence (2004)CrossRefzbMATHGoogle Scholar
  14. 14.
    Song, Y., Zhang, X.: Regularity of density for SDEs driven by degenerate Levy noises. Electron. J. Probab. 20(21), 1–27 (2015)zbMATHGoogle Scholar
  15. 15.
    Zhang, X.: Densities for SDEs driven by degenerate \(\alpha \)-stable processes. Ann. Probab. 42(5), 1885–1910 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, X.: Fundamental solutions of nonlocal Hörmander’s operators. Commun. Math. Stat. 4, 359–402 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhang, X.: Fundamental solutions of nonlocal Hörmander’s operators II. Ann. Probab. 45(3), 1799–1841 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsSun Yat-Sen UniversityGuangzhouPeople’s Republic of China
  2. 2.School of Statistics & Research Center of Applied StatisticsJiangxi University of Finance and EconomicsNanchangPeople’s Republic of China

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