General Existence Results for Reflected BSDE and BSDE

TL;DR

This paper establishes existence results for generalized reflected backward stochastic differential equations (GRBSDEs) and generalized BSDEs (GBSDEs) with continuous, growth, and quadratic conditions on the generator, applicable to bounded and unbounded terminal conditions and barriers.

Contribution

It provides new existence theorems for GRBSDEs and GBSDEs under broad growth conditions, extending previous results to more general settings.

Findings

Existence of solutions for GRBSDEs with continuous and quadratic growth generators.

Extension to unbounded terminal conditions and barriers.

Application potential in finance, control, and game theory.

Abstract

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator $fds + gdA_s$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. We deal with the case of a bounded terminal condition $\xi$ and a bounded barrier $L$ as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in \cite{EH}. The work is suggested by the interest the results might have in finance, control and game theory.