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The properties of value functions of time inhomogeneous optimal stoppingproblem and zero-sum game (Dynkin game) are studied through time dependentDirichlet form. Under the absolute continuity condition on the transitionfunction of the underlying diffusion process and some other assumptions, therefined solutions without exceptional starting points are proved to exist, andthe value functions of the optimal stopping and zero-sum game, which are finelyand cofinely continuous, are characterized as the solutions of some variationalinequalities, respectively.

Refined Solutions of Time Inhomogeneous Optimal Stopping Games via Dirichlet Form