Reflected Generalized Backward Doubly SDEs driven by L\'evy processes and Applications

TL;DR

This paper investigates reflected generalized backward doubly stochastic differential equations driven by Lévy processes, establishing existence and uniqueness, and providing a probabilistic representation for solutions to related stochastic partial differential integral equations with nonlinear boundary conditions.

Contribution

It introduces a new class of reflected backward doubly SDEs driven by Lévy processes and proves their well-posedness, extending the theory to include applications in SPDIEs with nonlinear Neumann boundary conditions.

Findings

Existence and uniqueness of solutions for RGBDSDELs with one continuous barrier.

Probabilistic representation for solutions to reflected SPDIEs.

Extension of backward doubly SDE theory to Lévy process-driven equations.

Abstract

In this paper, we study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with L\'evy process (RGBDSDELs, in short) with one continuous barrier. Under uniformly Lipschitz coefficients, we prove existence and uniqueness result by means of the penalization method and the fixed point theorem. As an application, this study allows us to give a probabilistic representation for the solutions to a class of reflected stochastic partial differential integral equations (SPDIEs, in short) with a nonlinear Neumann boundary condition.