Existence of optimal transition kernels (Q1084073)

A continuous-time stochastic control model is presented in which the controller is to select for every outcome a probability measure on the future of a canonical process after a fixed time, which minimizes an expected total cost criterion among all available measures for that outcome. Such an optimal transition kernel exists if the state space is Polish, the cost variable is Skorohod lower-semi-continuous and bounded below, and the admissibility multi-function is measurable. The theorem is applied to a problem in which one is to optimally choose the transition probabilities of a semi-Markov step process based on statistical estimates of the a priori sojourn time distributions.