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

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 assumptions 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 for reflected BSDEs 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 a penalization method and by some sequences of $L^{1}$ solutions of reflected BSDEs. The approach is based on a combination between existing methods, their refinement and perfection, but also on some novel ideas and techniques. These results strengthen some existing work on the $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs.