  In this paper, we consider a continuous-time Markov decision process (CTMDP)
in Borel spaces, where the certainty equivalent with respect to the exponential
utility of the total undiscounted cost is to be minimized. The cost rate is
nonnegative. We establish the optimality equation. Under the
compactness-continuity condition, we show the existence of a deterministic
stationary optimal policy. We reduce the risk-sensitive CTMDP problem to an
equivalent risk-sensitive discrete-time Markov decision process, which is with
the same state and action spaces as the original CTMDP. In particular, the
value iteration algorithm for the CTMDP problem follows from this reduction. We
do not need impose any condition on the growth of the transition and cost rate
in the state, and the controlled process could be explosive.

Optimization and Control