Continuous-Time Markov Decision Processes Under the Risk-Sensitive First Passage Discounted Cost Criterion

This paper studies the risk-sensitive first passage discounted cost criterion for continuous-time Markov decision processes with the Borel state and action spaces. The cost and transition rates are allowed to be unbounded. We introduce a new value iteration to establish the existence of a solution to the risk-sensitive first passage discounted cost optimality equation. Then applying the Feynman-Kac formula, we show that the risk-sensitive first passage discounted cost optimal value function is a unique solution to the risk-sensitive first passage discounted cost optimality equation. Moreover, we derive the existence of a deterministic Markov optimal policy in the class of randomized history-dependent policies. Finally, a cash flow model is given to illustrate the results.