Reflected backward stochastic differential equations with jumps in time-dependent random convex domains

TL;DR

This paper investigates multi-dimensional reflected backward stochastic differential equations with jumps within time-dependent convex domains, establishing existence, uniqueness, and approximation methods for solutions driven by Brownian motion and Poisson processes.

Contribution

It introduces a new framework for solving reflected backward stochastic differential equations with jumps in dynamic convex domains, including existence, uniqueness, and approximation results.

Findings

Proved existence and uniqueness of solutions.

Developed a penalization-based approximation method.

Extended the theory to time-dependent convex domains.

Abstract

In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a time-dependent adapted and continuous convex domain ${\cal{D}}=\{D_t, t\in[0,T]\}$. We prove the existence an uniqueness of the solution, and we also show that the solution of such equations may be approximated by backward stochastic differential equations with jumps reflected in appropriately defined discretizations of $\cal{D}$, via a penalization method.