# $L^1$ solutions to one-dimensional BSDEs with sublinear growth generators in $z$

## TL;DR

This paper establishes existence, comparison, and Lebesgue theorems for minimal $L^1$ solutions of one-dimensional BSDEs with generators exhibiting sublinear growth in $z$ and general growth in $y$, including discontinuous cases.

## Contribution

It introduces new existence and comparison results for $L^1$ solutions of BSDEs with sublinear $z$ growth, extending previous work to more general generator conditions.

## Key findings

- Existence of minimal $L^1$ solutions under broad conditions.
- Comparison theorems for $L^1$ solutions with various generator monotonicity.
- Extension to discontinuous generators in $y$.

## Abstract

This paper aims at solving a one-dimensional backward stochastic differential equation (BSDE for short) with only integrable parameters. We first establish the existence of a minimal $L^1$ solution for the BSDE when the generator $g$ is stronger continuous in $(y,z)$ and monotonic in $y$ as well as it has a general growth in $y$ and a sublinear growth in $z$. Particularly, the $g$ may be not uniformly continuous in $z$. Then, we put forward and prove a comparison theorem and a Levi type theorem on the minimal $L^1$ solutions. A Lebesgue type theorem on $L^1$ solutions is also obtained. Furthermore, we investigate the same problem in the case that $g$ may be discontinuous in $y$. Finally, we prove a general comparison theorem on $L^1$ solutions when $g$ is weakly monotonic in $y$ and uniformly continuous in $z$ as well as it has a stronger sublinear growth in $z$. As a byproduct, we also obtain a general existence and unique theorem on $L^1$ solutions. Our results extend some known works.