Bounded solutions, $L^p (p>1)$ solutions and $L^1$ solutions for one-dimensional BSDEs under general assumptions

TL;DR

This paper develops new existence, uniqueness, and comparison results for various solutions of one-dimensional BSDEs under weaker growth conditions on the generator, accommodating finite or infinite horizons and quadratic growth in $z$.

Contribution

It introduces broader conditions for BSDE solutions, relaxing traditional monotonicity assumptions and covering bounded, $L^p$, and $L^1$ solutions with general growth in $y$ and quadratic growth in $z$.

Findings

Established existence and uniqueness under weaker assumptions.

Extended results to infinite time horizons.

Improved upon previous results for $L^2$ solutions.

Abstract

This paper aims at solving one-dimensional backward stochastic differential equations (BSDEs) under weaker assumptions. We establish general existence, uniqueness, and comparison results for bounded solutions, $L^p (p>1)$ solutions and $L^1$ solutions of the BSDEs. The time horizon is allowed to be finite or infinite, and the generator $g$ is allowed to have a general growth in $y$ and a quadratic growth in $z$. As compensation, the generator $g$ needs to satisfy a kind of one-sided linear or super-linear growth condition in $y$, instead of the monotonicity condition in $y$ as is usually done. Many of our results improve virtually some known results, even though for the case of the finite time horizon and the case of the $L^2$ solution.