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 functionals and dynamics to be of polynomial growth. The problem is formulated as an extended doubly reflected BSDEs with a specific generator. show existence solution for this we prove saddle-point game. Moreover, in Markovian framework that value function unique viscosity associated Hamilton–Jacobi–Bellman (HJB) equation.