Generalized BSDE With 2-Reflecting Barriers and Stochastic Quadratic Growth

TL;DR

This paper establishes the existence of solutions for a generalized backward stochastic differential equation with two reflecting barriers, allowing for weaker assumptions and stochastic quadratic growth, with applications to game theory and finance.

Contribution

It introduces a new approach to solve GRBSDEs with weaker data assumptions and stochastic quadratic growth without P-integrability, expanding theoretical understanding.

Findings

Constructed a maximal solution under minimal assumptions.

Extended existence results to drivers with stochastic quadratic growth.

Applicable to Dynkin games and American options.

Abstract

We study the existence of a solution for a one-dimensional generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under assumptions on the input data which are weaker than that on the current literature. In particular, we construct a maximal solution for such a GRBSDE when the terminal condition \xi is only F_T-measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z without assuming any P-integrability conditions. The work is suggested by the interest the results might have in Dynkin game problem and American game option.