Discounted Continuous-time Markov Decision Processes with Unbounded   Rates: the Dynamic Programming Approach
  This paper deals with unconstrained discounted continuous-time Markov decision processes in Borel state and action spaces. Under some conditions imposed on the primitives, allowing unbounded transition rates and unbounded (from both above and below) cost rates, we show the regularity of the controlled process, which ensures the underlying models to be well defined. Then we develop the dynamic programming approach by showing that the Bellman equation is satisfied (by the optimal value). Finally, under some compactness-continuity conditions, we obtain the existence of a deterministic stationary optimal policy out of the class of randomized history-dependent policies. 
Alexey Piunovskiy and Yi Zhang
math.OC