$L^1$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions

TL;DR

This paper establishes existence, uniqueness, and approximation results for $L^1$ solutions of non-reflected and reflected BSDEs with one or two barriers under general conditions, extending previous work in the field.

Contribution

It introduces a generalized Mokobodzki condition necessary for solutions and demonstrates approximation of solutions via penalization and sequences, broadening the theoretical framework.

Findings

Generalized Mokobodzki condition is necessary for existence

Solutions can be approximated by penalization methods

Extends existing results on $L^1$ solutions of BSDEs

Abstract

We establish several existence, uniqueness and comparison results for $L^1$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under the assumption that the generator $g$ satisfies a one-sided Osgood condition together with a very general growth condition in $y$, a uniform continuity condition and/or a sub-linear growth condition in $z$, and a generalized Mokobodzki condition which relates the growth of $g$ and that of the barriers. This generalized Mokobodzki condition is proved to be necessary for existence of $L^1$ solutions of the reflected BSDEs. We also prove that the $L^1$ solutions of reflected BSDEs can be approximated by the penalization method and by some sequences of $L^1$ solutions of reflected BSDEs. These results strengthen some existing work on the $L^1$ solutions of non-reflected BSDEs and reflected BSDEs.