  In this paper, we consider the discounted continuous-time Markov decision
process (CTMDP) with a lower bounding function. In this model, the negative
part of each cost rate is bounded by the drift function, say $w$, whereas the
positive part is allowed to be arbitrarily unbounded. Our focus is on the
existence of a stationary optimal policy for the discounted CTMDP problems out
of the more general class. Both constrained and unconstrained problems are
considered. Our investigations are based on a useful transformation for
nonhomogeneous Markov pure jump processes that has not yet been widely applied
to the study of CTMDPs. This technique was not employed in previous literature,
but it clarifies the roles of the imposed conditions in a rather transparent
way. As a consequence, we withdraw and weaken several conditions commonly
imposed in the literature.
