$L^p\ (p>1)$ solutions of BSDEs with generators satisfying some non-uniform conditions in $t$ and $\omega$

TL;DR

This paper establishes existence, uniqueness, and comparison results for $L^p$ solutions of one-dimensional BSDEs with generators under non-uniform conditions in time and probability, extending previous work in the field.

Contribution

It introduces new existence and uniqueness theorems for BSDEs with non-uniform generator conditions, improving upon existing results for $L^p$ solutions.

Findings

Proved existence and uniqueness of solutions under non-uniform conditions.

Established a comparison theorem for these BSDEs.

Provided an existence result for minimal solutions.

Abstract

This paper is devoted to the $L^p$ ($p>1$) solutions of one-dimensional backward stochastic differential equations (BSDEs for short) with general time intervals and generators satisfying some non-uniform conditions in $t$ and $\omega$. An existence and uniqueness result, a comparison theorem and an existence result for the minimal solutions are respectively obtained, which considerably improve some known works. Some classical techniques used to deal with the existence and uniqueness of $L^p$ ($p>1$) solutions of BSDEs with Lipschitz or linear-growth generators are also developed in this paper.