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

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 Itô'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.

Received February 16, 2022; accepted April 15, 2023. arXiv admin note: text overlap with arXiv:1812.07990