Reflected Backward SDEs with General Jumps

TL;DR

This paper establishes existence and uniqueness results for one-dimensional reflected backward stochastic differential equations driven by Brownian motion and Poisson jumps, including cases with two barriers, using penalization, Snell envelope, and fixed point methods.

Contribution

It provides the first comprehensive solution framework for reflected BSDEs with general jumps and two barriers, extending previous results to more complex jump structures.

Findings

Proved existence and uniqueness for reflected BSDEs with arbitrary jumps.

Extended results to BSDEs with two reflecting barriers.

Developed methods combining penalization, Snell envelope, and fixed point techniques.

Abstract

In the first part of this paper we give a solution for the one-dimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. The reflecting process is right continuous with left limits (rcll for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the result in the general framework we use a fixed point argument in an appropriate space. The second part of the paper is related to BSDEs with two reflecting barriers. Once more we prove the existence and uniqueness of the solution of the BSDE.