Sufficiency of Markov Policies for Continuous-Time Jump Markov Decision Processes

One of the basic facts known for discrete-time Markov decision processes is that, if the probability distribution of an initial state is fixed, then for every policy it is easy to construct a (randomized) Markov policy with the same marginal distributions of state-action pairs as for the original policy. This equality of marginal distributions implies that the values of major objective criteria, including expected discounted total costs and average rewards per unit time, are equal for these two policies. This paper investigates the validity of the similar fact for continuous-time jump Markov decision processes (CTJMDPs). It is shown in this paper that the equality of marginal distributions takes place for a CTJMDP if the corresponding Markov policy defines a nonexplosive jump Markov process. If this Markov process is explosive, then at each time instance, the marginal probability, that a state-action pair belongs to a measurable set of state-action pairs, is not greater for the described Markov policy than the same probability for the original policy. These results are applied in this paper to CTJMDPs with expected discounted total costs and average costs per unit time. It is shown for these criteria that, if the initial state distribution is fixed, then for every policy, there exists a Markov policy with the same or better value of the objective function.