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In this paper we study the nonzero-sum Dynkin game in continuous time which is a two player non-cooperative game on stopping times. We show that it has a Nash equilibrium point for general stochastic processes. As an application, we consider the problem of pricing American game contingent claims by the utility maximization approach. AMS Classification subjects: 91A15; 91A10; 91A30; 60G40; 91A60.doi:10.1137/080738933 fatcat:lrngn7d4knbhpmo3cf6zt4s5x4