Existence of an optimal control for stochastic systems governed by ito equations

We give existence theorems for stochastic control problems with a lower semicontinuous cost functional and governed by Ito equations. We prove that two formulations of the fundamental problem are equivalent, one involving nonanticipative controls and the other involving (measurable) feedback controls. We then use the concept ofconvergence in distribution to prove existence for the first problem, and hence for the second as well. While our work has certain similarities with a paper of Kushner, our techniques are different and lead to more general results.