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

TL;DR

This paper investigates backward doubly stochastic differential equations with jumps driven by Brownian motion and Poisson processes, establishing existence, uniqueness, and continuous dependence results under non-Lipschitz conditions, and interprets solutions to related stochastic partial differential-integral equations.

Contribution

It provides the first comprehensive analysis of BDSDEPs with jumps under non-Lipschitz conditions and links their solutions to quasilinear stochastic partial differential-integral equations.

Findings

Existence and uniqueness of solutions under non-Lipschitz conditions.

Continuous dependence of solutions on parameters.

Probabilistic interpretation of solutions to SPDIEs.

Abstract

In this paper, we study backward doubly stochastic differential equations driven by Brownian motions and Poisson process (BDSDEP in short) with non-Lipschitz coefficients on random time interval. The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) is tthe solutionBDSDEP. Under non-Lipschitz conditions, the existence and uniqueness results for measurable solutions of BDSDEP are established via the smoothing technique. Then, the continuous dependence for solutions of BDSDEP is derived. Finally, the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.