Continuous-time Markov decision processes with exponential utility

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 essentially do not need to impose a condition on the growth of the transition and cost rate in the state, and the controlled process could be explosive.