The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps

Abstract

Consider jump-type stochastic differential equations with the drift, diffusion and jump terms. Logarithmic derivatives of densities for the solution process are studied, and the Bismut-Elworthy-Li type formulae can be 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 Markovian property of the process.

29 pages, to appear in Journal of Theoretical Probability

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• Bismut Formula for Lions Derivative of Distribution-Path Dependent SDEs
• Partial Uncertainty and Applications to Risk-Averse Valuation
• Gradient estimates for semigroups associated with stochastic differential equations driven by cylindrical Lévy processes
• Joint distributions for stochastic functional differential equations