Abstract
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.
Highlights
The nonlinear backward stochastic differential equations BSDEs in short had been introduced by Pardoux and Peng 1, who proved the existence and uniqueness of adapted solutions under suitable assumptions
We will study the existence of the stochastic recursive zero-sum differential game problem using the result of BSDEs
From the result in 12, the stochastic mixed zero-sum differential game problem is possibly connected with BSDEs with two reflecting barriers
Summary
The nonlinear backward stochastic differential equations BSDEs in short had been introduced by Pardoux and Peng 1 , who proved the existence and uniqueness of adapted solutions under suitable assumptions. 2, we proved the existence of a saddle point for the stochastic recursive zero-sum differential game problem and got the optimal payoff function by the solution of one specific BSDE. Using the result of reflected BSDEs with two barriers, we got the saddle point and optimal stopping strategy for the recursive mixed game problem which has more general significance than that in 9.
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