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Under the notable Issacs's condition on the Hamiltonian, the existence results of a saddle point are obtained for the stochastic recursive zero-sum differential game and mixed differential game problem, that is, the agents can also decide the optimal stopping time. The main tools are backward stochastic differential equations (BSDEs) and double-barrier reflected BSDEs. As the motivation and application background, when loan interest rate is higher than the deposit one, the American game option pricing problem can be formulated to stochastic recursive mixed zero-sum differential game problem. One example with explicit optimal solution of the saddle point is also given to illustrate the theoretical results

Stochastic Recursive Zero-Sum Differential Game and Mixed Zero-Sum Differential Game Problem