Reflections on BSDEs

TL;DR

This paper establishes well-posedness for a broad class of backward stochastic differential equations, including reflected variants with optional obstacles, unifying discrete and continuous cases, and providing conditions for comparison principles.

Contribution

It offers a unified framework for well-posedness of BSDEs with weighted L^2 data, including discontinuous generators, and sharp results that cannot be improved within this framework.

Findings

Proved well-posedness for BSDEs with discontinuous generators.

Unified treatment of discrete-time and continuous-time BSDEs.

Provided conditions for comparison principles.

Abstract

We prove well-posedness results for backward stochastic differential equations (BSDEs) and reflected BSDEs with an optional obstacle process in the case of appropriately weighted $\mathbb{L}^2$-data when the generator is integrated with respect to a possibly purely discontinuous process. This leads to a unified treatment of discrete-time and continuous-time (reflected) BSDEs. We compare our well-posedness results with the current literature and highlight that our results are sharp and cannot be improved within the framework presented here. Finally, we provide sufficient conditions for a comparison principle.