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We investigate a two-player zero-sum stochastic differential game problemwith the state process being constrained in a connected bounded closed domain,and the cost functional described by the solution of a generalized backwardstochastic differential equation (GBSDE for short). We show that the valuefunctions enjoy a (strong) dynamic programming principle, and are the uniqueviscosity solution of the associated Hamilton-Jacobi-Bellman-Isaacs equationswith nonlinear Neumann boundary problems. To obtain the existence for viscositysolutions, we provide a new approach utilizing the representation theorem forgenerators of the GBSDE, which is proved by a random time change method and isa novel result in its own right.

Stochastic differential games with state constraints and Isaacs equations with nonlinear Neumann problems