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Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps

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  • Ishikawa, Yasushi
  • Kunita, Hiroshi

Abstract

We study the existence and smoothness of densities of laws of solutions of a canonical stochastic differential equation (SDE) driven by a Lévy process through the Malliavin calculus on the Wiener-Poisson space. Our assumption needed for the equation is very simple, since we are considering the canonical SDE. Assuming that the Lévy process is nondegenerate, we prove the existence of a smooth density in the case where the coefficients of the equation are nondegenerate. Our main result is stated in Theorem 1.1.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:12:p:1743-1769
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    Citations

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    Cited by:

    1. Masafumi Hayashi, 2010. "Coefficients of Asymptotic Expansions of SDE with Jumps," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 17(4), pages 373-389, December.
    2. 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.
    3. Xin Guo & Robert A. Jarrow & Yan Zeng, 2009. "Credit Risk Models with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 320-332, May.
    4. Andrey Borisov & Andrey Gorshenin, 2022. "Identification of Continuous-Discrete Hidden Markov Models with Multiplicative Observation Noise," Mathematics, MDPI, vol. 10(17), pages 1-20, August.
    5. Andrey Borisov & Alexey Bosov & Gregory Miller & Igor Sokolov, 2021. "Partial Diffusion Markov Model of Heterogeneous TCP Link: Optimization with Incomplete Information," Mathematics, MDPI, vol. 9(14), pages 1-31, July.
    6. Alexander Steinicke, 2016. "Functionals of a Lévy Process on Canonical and Generic Probability Spaces," Journal of Theoretical Probability, Springer, vol. 29(2), pages 443-458, June.
    7. Clément, Emmanuelle & Gloter, Arnaud, 2015. "Local Asymptotic Mixed Normality property for discretely observed stochastic differential equations driven by stable Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2316-2352.
    8. Goldys, Beniamin & Wu, Wei, 2019. "On a class of singular stochastic control problems driven by Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3174-3206.
    9. Atsushi Takeuchi, 2010. "Bismut–Elworthy–Li-Type Formulae for Stochastic Differential Equations with Jumps," Journal of Theoretical Probability, Springer, vol. 23(2), pages 576-604, June.
    10. 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.
    11. Schmisser, Émeline, 2014. "Non-parametric adaptive estimation of the drift for a jump diffusion process," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 883-914.
    12. 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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