On stochastic relaxed control for partially observed diffusions

In this paper we are concerned with stochastic relaxed control problems of the following kind. Let X(t), t ≥ 0, denote the state of a process being controlled, Y(t), t ≥ 0, the observation process and p(t, ·) a relaxed control, that is a process with values probability measures on the control region Г. The state and observation processes are governed by stochastic differential equations and where B and W are independent Brownian motions with values in Rn and Rm respectively, (put m = 1 for simplicity).