A Generalized Mixed Zero-sum Stochastic Differential Game and Double Barrier Reflected BSDEs with Quadratic Growth Coefficient

TL;DR

This paper extends the theory of stochastic differential games by analyzing mixed zero-sum games with doubly controlled reflected backward stochastic equations featuring quadratic growth coefficients, establishing their value functions as viscosity solutions.

Contribution

It generalizes risk-sensitive payoffs to a broader class of stochastic differential games with reflection and quadratic growth, proving the determinism and uniqueness of value functions.

Findings

Value functions are deterministic.

Value functions are unique viscosity solutions.

The framework generalizes risk-sensitive payoffs.

Abstract

This article is dedicated to the study of mixed zero-sum two-player stochastic differential games in the situation when the player's cost functionals are modeled by doubly controlled reflected backward stochastic equations with two barriers whose coefficients have quadratic growth in Z. This is a generalization of the risk-sensitive payoffs. We show that the lower and the upper value function associated with this stochastic differential game with reflection are deterministic and they are also the unique viscosity solutions for two Isaacs equations with obstacles.