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Smooth density and its short time estimate for jump process determined by SDE

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  • Ishikawa, Yasushi
  • Kunita, Hiroshi
  • Tsuchiya, Masaaki

Abstract

We study a nondegenerate jump process on Euclidean space determined by SDE. We show the existence of the smooth density p(s,x;t,y) of its transition probability and its short time asymptotics as t−s→0. Assumptions required for these facts are relaxed considerably from past works by Picard and Ishikawa–Kunita. We show these facts using Malliavin calculus on Poisson space. Our calculus is simpler and more efficient than previous works.

Suggested Citation

  • Ishikawa, Yasushi & Kunita, Hiroshi & Tsuchiya, Masaaki, 2018. "Smooth density and its short time estimate for jump process determined by SDE," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 3181-3219.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:9:p:3181-3219
    DOI: 10.1016/j.spa.2017.10.016
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    References listed on IDEAS

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    1. Kunita, Hiroshi, 2010. "Itô's stochastic calculus: Its surprising power for applications," Stochastic Processes and their Applications, Elsevier, vol. 120(5), pages 622-652, May.
    2. Picard, Jean, 1997. "Density in small time at accessible points for jump processes," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 251-279, May.
    3. Ishikawa, Yasushi & Kunita, Hiroshi, 2006. "Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1743-1769, December.
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    Cited by:

    1. 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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