  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.
