Dynkin games for Markov processes associated with semi-Dirichlet forms

We consider Dynkin games for Markov processes associated with semi-Dirichlet forms. Dynkin games are the optimal stopping games introduced as the models of zero-sum games by two players. We prove that the solution to the certain variational inequality with two obstacles is the equilibrium price of the Dynkin game. Moreover, we obtain the saddle point of the game.