  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.
