The Transformation Method for Continuous-Time Markov Decision Processes

In this paper, we show that a discounted continuous-time Markov decision process in Borel spaces with randomized history-dependent policies, arbitrarily unbounded transition rates and a non-negative reward rate is equivalent to a discrete-time Markov decision process. Based on a completely new proof, which does not involve Kolmogorov's forward equation, it is shown that the value function for both models is given by the minimal non-negative solution to the same Bellman equation. A verifiable necessary and sufficient condition for the finiteness of this value function is given, which induces a new condition for the non-explosion of the underlying controlled process.