Skip to main content
SpringerLink
Log in

Density functions of doubly-perturbed stochastic differential equations with jumps

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

We consider a real-valued doubly-perturbed stochastic differential equation driven by a subordinated Brownian motion. By using classic Malliavin calculus, we prove that the law of the solution is absolutely continuous with respect to the Lebesgue measure on ℝ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Chaumont L, Doney R A. Some calculations for doubly-perturbed Brownian motion. Stochastic Process Appl, 2000, 85: 61–74

    Article  MathSciNet  MATH  Google Scholar 

  2. Davis B. Weak limits of perturbed random walks and the equation Y t = B t+ αsups≤t Y s +ß infs≤t Y s. Ann Probab, 1996, 24: 2007–2023

    Article  MathSciNet  Google Scholar 

  3. Davis B. Brownian motion and random walk perturbed at extrema. Probab Theory Related Fields, 1999, 113: 501–518

    Article  MathSciNet  MATH  Google Scholar 

  4. Doney R A. Some calculations for perturbed Brownian motion. In: Azéma J, Émery M, Ledoux M, Yor M, eds. Séminaire de Probabilités XXXII. Lecture Notes in Math, Vol 1686. Berlin: Springer, 1998, 231–236

    Google Scholar 

  5. Doney R A, Zhang T S. Perturbed Skorohod equation and perturbed reflected diffusion processes. Ann Inst Henri Poincaré Probab Stat, 2005, 41: 107–121

    Article  MathSciNet  MATH  Google Scholar 

  6. Kusuoka S. Malliavin calculus for stochastic differential equations driven by subordinated Brownian motions. Kyoto J Math, 2009, 50: 491–520

    Article  MathSciNet  MATH  Google Scholar 

  7. Le Gall J F, Yor M. Excursions browniennes et carrés de processus de Bessel. C R Acad Sci Sr 1, Math, 1986, 303: 73–76

    MATH  Google Scholar 

  8. Luo J.W. Doubly perturbed jump-diffusion processes. J Math Anal Appl, 2009, 351: 147–151

    Article  MathSciNet  MATH  Google Scholar 

  9. Nualart D. The Malliavin Calculus and Related Topics. New York: Springer-Verlag, 2006

    MATH  Google Scholar 

  10. Perman M, Werner W. Perturbed Brownian motion. Probab Theory Related Fields, 1997, 108: 357–383

    Article  MathSciNet  MATH  Google Scholar 

  11. Sato K. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge Univ Press, 1999

    MATH  Google Scholar 

  12. Werner W. Some remarks on perturbed Brownian motion. In: Azema J, Emery M, Meyer P A, Yor M, eds. Seminaire de Probabilités XXIX. Lecture Notes in Math, Vol 1613. Berlin: Springer, 1995, 37–43

    Google Scholar 

  13. Yue W, Zhang T S. Absolutely continuous of the laws of perturbed diffusion processes and perturbed reflected diffusion processes. J Theoret Probab, 2015, 28: 587–618

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang X C. Densities for SDEs driven by degenerated-stable processes. Ann Probab, 2014, 42: 1885–1910

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11501286) and the Natural Science Foundation of Jiangsu Province (No. BK20150564).

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Yulin Song

Authors
  1. Yulin Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yulin Song.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Song, Y. Density functions of doubly-perturbed stochastic differential equations with jumps. Front. Math. China 13, 161–172 (2018). https://doi.org/10.1007/s11464-017-0659-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-017-0659-7

Keywords

MSC

Search

Navigation