Backward stochastic differential equations with unbounded generators

TL;DR

This paper investigates two classes of backward stochastic differential equations with unbounded generators, establishing existence, uniqueness, and comparison results under new conditions, thus broadening the scope of solvable equations.

Contribution

It introduces new sufficient conditions for existence and uniqueness of solutions for BSDEs with unbounded generators, extending previous results.

Findings

Established existence and uniqueness under Lipschitz-type conditions.

Proved a comparison theorem for the equations.

Extended solvability to more general unbounded generators.

Abstract

In this paper we consider two classes of backward stochastic differential equations. Firstly, under a Lipschitz-type condition on the generator of the equation, which can also be unbounded, we give sufficient conditions for the existence of a unique solution pair. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Secondly, under the linear growth and continuity assumptions on the possibly unbounded generator, we prove the existence of the solution pair. This class of equations is more general than the existing ones.