Bismut-Elworthy-Li Formulae for Forward-Backward SDEs with Jumps and Applications

TL;DR

This paper derives Bismut-Elworthy-Li type formulas for forward-backward SDEs with jumps using the lent particle method, enabling analysis of nonlocal PDEs with minimal regularity assumptions.

Contribution

It introduces a novel derivative formula for FBSDEs with jumps using the lent particle method, not previously established in the literature.

Findings

Established derivative formulas for FBSDEs with jumps.

Proved existence and uniqueness of solutions for certain nonlocal PDEs.

Extended applicability to PDEs with irregular initial data and coefficients.

Abstract

Under nondegeneracy assumptions on the diffusion coefficients, we establish the derivative formulae of Bismut-Elworthy-Li's type for forward-backward stochastic differential equations with respect to Poisson random measure using the lent particle method created by Bouleau and Denis, which is not given before. Applying this formula, the existence and uniqueness of a solution of nonlocal quasi-linear integral partial differential equations, which are differentiable with respect to the space variable, are obtained, even if the initial datum and coefficients of this equation are not.