Reflected BSDEs on Filtered Probability Spaces

TL;DR

This paper investigates the existence and uniqueness of solutions to reflected backward stochastic differential equations with irregular barriers on general filtered probability spaces, extending results to various integrability and filtration conditions.

Contribution

It provides new existence and uniqueness results for reflected BSDEs with irregular barriers under broad conditions, including general filtrations and different integrability regimes.

Findings

Established existence and uniqueness for $p=1$ with right-continuous and complete filtrations.

Extended results to $p ext{ in }(1,2]$ with quasi-left continuous filtrations.

Applied findings to Markov-type reflected backward equations driven by Hunt processes.

Abstract

We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, $L^p$ data and generators satisfying weak integrability conditions. We deal with equations on general filtered probability spaces. In case the generator does not depend on the $z$ variable, we first consider the case $p=1$ and we only assume that the underlying filtration satisfies the usual conditions of right-continuity and completeness. Additional integrability properties of solutions are established if $p\in(1,2]$ and the filtration is quasi-left continuous. In case the generator depends on $z$, we assume that $p=2$, the filtration satisfies the usual conditions and additionally that it is separable. Our results apply for instance to Markov-type reflected backward equations driven by general Hunt processes.