Mixed Zero-Sum Stochastic Differential Game and Doubly Reflected BSDEs with a Specific Generator

TL;DR

This paper investigates a complex stochastic differential game involving polynomial growth functionals, establishing existence of solutions for extended doubly reflected BSDEs and linking the value function to a unique viscosity solution of the HJB equation.

Contribution

It introduces a novel formulation of the game as extended doubly reflected BSDEs with a specific generator and proves existence of solutions and saddle-points.

Findings

Existence of solutions for the extended doubly reflected BSDEs

Existence of a saddle-point in the game

Value function as a unique viscosity solution of the HJB equation

Abstract

This paper studies the mixed zero-sum stochastic differential game problem. We allow the functionals and dynamics to be of polynomial growth. The problem is formulated as an extended doubly reflected BSDEs with a specific generator. We show the existence of solution for this doubly reflected BSDEs and we prove the existence of a saddle-point of the game. Moreover, in the Markovian framework we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.