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Smooth densities for solutions to stochastic differential equations with jumps

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  • Cass, Thomas

Abstract

We consider a solution xt to a generic Markovian jump diffusion and show that for any t0>0 the law of xt0 has a C[infinity] density with respect to the Lebesgue measure under a uniform version of the Hörmander conditions. Unlike previous results in the area the result covers a class of infinite activity jump processes. The result is accomplished using carefully crafted refinements to the classical arguments used in proving the smoothness of density via Malliavin calculus. In particular, we provide a proof that the semimartingale inequality of J. Norris persists for discontinuous semimartingales when the jumps are small.

Suggested Citation

  • Cass, Thomas, 2009. "Smooth densities for solutions to stochastic differential equations with jumps," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1416-1435, May.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:5:p:1416-1435
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    Cited by:

    1. Iguchi, Yuga & Beskos, Alexandros & Graham, Matthew M., 2024. "Parameter inference for degenerate diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    2. Kenichiro Shiraya & Akihiko Takahashi, 2019. "Pricing Average and Spread Options Under Local-Stochastic Volatility Jump-Diffusion Models," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 303-333, February.
    3. Giesecke, Kay & Schwenkler, Gustavo, 2018. "Filtered likelihood for point processes," Journal of Econometrics, Elsevier, vol. 204(1), pages 33-53.
    4. Guay, François & Schwenkler, Gustavo, 2021. "Efficient estimation and filtering for multivariate jump–diffusions," Journal of Econometrics, Elsevier, vol. 223(1), pages 251-275.
    5. Marino, L. & Menozzi, S., 2023. "Weak well-posedness for a class of degenerate Lévy-driven SDEs with Hölder continuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 106-170.
    6. Christian Fries & Joerg Kampen, 2010. "Global existence, regularity and a probabilistic scheme for a class of ultraparabolic Cauchy problems," Papers 1002.5031, arXiv.org, revised Oct 2012.
    7. Zimo Hao & Xuhui Peng & Xicheng Zhang, 2021. "Hörmander’s Hypoelliptic Theorem for Nonlocal Operators," Journal of Theoretical Probability, Springer, vol. 34(4), pages 1870-1916, December.

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