Continuous-time Zero-Sum Games for Markov decision processes under the risk-sensitive first passage discounted cost criterion

We analyze zero-sum stochastic games for controlled continuous-time Markov decision processes (CTMDPs) over a general state space, employing the risk- sensitive first-passage discounted cost criterion. Both transition and cost rates may be unbounded. Assuming stability, we prove the existence and uniqueness of the solution to the Hamilton-Jacobi-Isaacs (HJI) equation through a value iteration approach. By applying the Feynman-Kac formula, we confirm the presence of a saddle-point equilibrium among Markov strategies, describing it via the associated HJI equation. Lastly, we validate our findings and assumptions with numerical examples.