Policy and Value Iteration Algorithms for Risk-Sensitive First Exit Time Zero-Sum Games with Varying Discount Factors

The risk-sensitive first exit time stochastic zero-sum game problem for pure jump processes on a general state space with state-dependent discount factors is analysed. Under minimal assumptions, we prove the existence and uniqueness of the value of the game over the history-dependent policy space for bounded cost function and transition rates. We propose a value iteration algorithm and establish its convergence on a general state space to approximate the value of the game. For countable state space, we propose a new policy iteration algorithm and prove its convergence to the Saddle point equilibrium. Finally, we present two examples with bounded transition and cost rates to illustrate our convergence results corresponding to the general Borel and countable state space settings.