Continuity of cost functionals in diffusion processes and its application to an existence theorem of optimal controls

A cost functional, which is the integral of a cost rate over a random time, is associated with a control system described by a stochastic differential equation. The control policy is confined to measurable Markov controls. In connection to the problem of selecting a minimizing control, two results are shown in this paper. The first one is the continuity dependence of the cost functional on controls, i.e., it is shown that if a sequence of controls converges weakly to a control then the corresponding sequence of the cost functionals converges to the cost functional corresponding to the limiting control. Using this result, the second one is to show an existence theorem of optimal stochastic controls among an appropriate set of admissible controls under some assumptions.