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

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.