# Existence, uniqueness and stability of $L^1$ solutions for multidimensional BSDEs with generators of one-sided Osgood type

## TL;DR

This paper proves existence, uniqueness, and stability of $L^1$ solutions for multidimensional BSDEs with generators satisfying one-sided Osgood and growth conditions, advancing theoretical understanding in stochastic differential equations.

## Contribution

It introduces a comprehensive existence and uniqueness framework for $L^1$ solutions under new conditions and establishes the first stability theorem for these solutions.

## Key findings

- Proved existence and uniqueness of $L^1$ solutions under Osgood and growth conditions.
- Established the first stability theorem for $L^1$ solutions.
- Investigated a new type of $L^1$ solution.

## Abstract

We establish a general existence and uniqueness result of $L^1$ solution for a multidimensional backward stochastic differential equation (BSDE for short) with generator $g$ satisfying a one-sided Osgood condition as well as a general growth condition in $y$, and a Lipschitz condition together with a sublinear growth condition in $z$, which improves some existing results. In particular, we put forward and prove a stability theorem of the $L^1$ solutions for the first time. A new type of $L^1$ solution is also investigated. Some delicate techniques involved in the relationship between convergence in $L^1$ and in probability and dividing appropriately the time interval play crucial roles in our proofs.