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

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