# $L^p$ solutions of doubly reflected BSDEs under general assumptions

## TL;DR

This paper establishes existence, uniqueness, and approximation results for $L^p$ solutions of doubly reflected backward stochastic differential equations under a generalized Mokobodzki condition, broadening the understanding of their solvability.

## Contribution

It introduces a generalized Mokobodzki condition for doubly reflected BSDEs and proves its necessity and sufficiency for $L^p$ solutions, along with approximation methods.

## Key findings

- Existence and uniqueness of $L^p$ solutions under generalized Mokobodzki condition.
- Necessity of the Mokobodzki condition for solution existence.
- Solutions can be approximated via penalization and solution sequences.

## Abstract

Under a generalized Mokobodzki condition for reflected BSDEs with two continuous barriers which relates the growth of the generator $g$ and that of the barriers, we establish several existence and uniqueness results on $L^p\ (p>1)$ solutions of doubly reflected BSDEs with generators satisfying a one-sided Osgood condition together with a general growth in the state variable $y$, and a uniform continuity condition or a linear growth condition in the state variable $z$. This Mokobodzki condition is also proved to be necessary for existence of the $L^p$ solutions. And, we prove that the $L^p$ solutions can be approximated by the penalization method and by some sequences of the $L^p$ solutions of doubly reflected BSDEs.