First passage risk probability optimality for partially observable discrete-time zero-sum games

In this paper we study partially observable discrete-time zero-sum games with Borel state spaces and unbounded reward functions. The optimality criterion is the first passage risk probability criterion. We first introduce a new auxiliary zero-sum game problem and establish the corresponding Shapley equation. Then via an approximation approach we obtain the existence of a solution to the Shapley equation under the first passage risk probability criterion. Moreover, applying the Shapley equation we prove the existence of a saddle-point equilibrium.