Forward-Backward Doubly Stochastic Differential Equations with Random Jumps and Stochastic Partial Differential-Integral Equations

TL;DR

This paper introduces a class of forward-backward doubly stochastic differential equations driven by Brownian motions and Poisson processes, providing existence, uniqueness, and regularity results, and linking them to stochastic partial differential-integral equations and stochastic control.

Contribution

It develops new theoretical results for FBDSDEP with jumps, including existence, uniqueness, and parameter dependence, and offers a probabilistic interpretation for related SPDIEs.

Findings

Established existence and uniqueness of solutions under monotonicity conditions.

Proved continuity and differentiability of solutions with respect to parameters.

Provided a probabilistic representation for solutions to certain SPDIEs.

Abstract

In this paper, we study forward-backward doubly stochastic differential equations driven by Brownian motions and Poisson process (FBDSDEP in short). Both the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) and stochastic Hamiltonian systems arising in stochastic optimal control problems with random jumps are treated with FBDSDEP. Under some monotonicity assumptions, the existence and uniqueness results for measurable solutions of FBDSDEP are established via a method of continuation. Furthermore, the continuity and differentiability of the solutions of FBDSDEP depending on parameters is discussed. Finally, the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.