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

TL;DR

This paper derives Bismut-Elworthy-Li type formulas for jump-type stochastic differential equations, providing a way to compute derivatives of densities using the process's Markov property under elliptic conditions.

Contribution

It introduces a novel approach to obtain derivative formulas for jump SDEs using the Kolmogorov backward equation and Markovian properties.

Findings

Derived Bismut-Elworthy-Li formulas for jump SDEs

Applicable under uniform ellipticity conditions

Utilizes Kolmogorov backward equation approach

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.