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Bismut–Elworthy–Li-Type Formulae for Stochastic Differential Equations with Jumps

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  • Atsushi Takeuchi

    (Osaka City University)

Abstract

We consider jump-type stochastic differential equations with drift, diffusion, and jump terms. Logarithmic derivatives of densities for the solution process are studied, and Bismut–Elworthy–Li-type formulae are obtained under the uniformly elliptic condition on the coefficients of the diffusion and jump terms. Our approach is based upon the Kolmogorov backward equation by making full use of the Markov property of the process.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:2:d:10.1007_s10959-010-0280-0
    DOI: 10.1007/s10959-010-0280-0
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    References listed on IDEAS

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    1. Youssef El-Khatib & Nicolas Privault, 2004. "Computations of Greeks in a market with jumps via the Malliavin calculus," Finance and Stochastics, Springer, vol. 8(2), pages 161-179, May.
    2. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    3. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    4. 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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