Existence and uniqueness for reflected BSDE with multivariate point process and right upper-semi-continuous obstacle

TL;DR

This paper establishes the existence and uniqueness of solutions for reflected backward stochastic differential equations driven by multivariate point processes with right upper-semicontinuous obstacles, extending previous results to more general obstacle processes.

Contribution

It introduces a novel framework for reflected BSDEs with non-right-continuous obstacles driven by multivariate point processes, employing advanced decomposition and optimal stopping techniques.

Findings

Proved existence and uniqueness of solutions under new conditions.

Developed a comparison theorem for these reflected BSDEs.

Extended the theory to obstacles that are right upper-semicontinuous but not necessarily right-continuous.

Abstract

In a noise driving by a multivariate point process $\mu$ with predictable compensator $\nu$, we prove existence and uniqueness of the reflected backward stochastic differential equation's solution with a lower obstacle $(\xi_{t})_{t\in[0,T]}$ which is assumed to be right upper-semicontinuous but not necessarily right-continuous process and a Lipschitz driver $f$. The result is established by using Mertens decomposition of optional strong (but not necessarily right continuous) super-martingales, an appropriate generalization of It\^{o}'s formula due to Gal'chouk and Lenglart and some tools from optimal stopping theory. A comparison theorem for this type of equations is given.