One dimensional reflected BSDEs with two barriers under logarithmic growth and applications

TL;DR

This paper establishes existence and uniqueness results for one-dimensional reflected backward stochastic differential equations with two barriers under logarithmic growth conditions, and applies these results to stochastic games and PDEs.

Contribution

It introduces a method to prove existence and uniqueness for reflected BSDEs with logarithmic growth, expanding their applicability to stochastic games and PDEs.

Findings

Existence and uniqueness of solutions under logarithmic growth conditions.

Broadened class of functions for stochastic differential games.

Unique viscosity solutions for related double obstacle PDEs.

Abstract

In this paper we deal with the problem of the existence and the uniqueness of a solution for one dimensional reflected backward stochastic differential equations with two strictly separated barriers when the generator is allowing a logarithmic growth $(|y||\ln|y||+|z|\sqrt{|\ln|z||})$ in the state variables $y$ and $z$. The terminal value $\xi$ and the obstacle processes $(L_t)_{0\leq t\leq T}$ and $(U_t)_{0\leq t\leq T}$ are $L^p$-integrable for a suitable $p > 2$. The main idea is to use the concept of local solution to construct the global one. As applications, we broaden the class of functions for which mixed zero-sum stochastic differential games admit an optimal strategy and the related double obstacle partial differential equation problem has a unique viscosity solution.