Approximation in optimal control of diffusion processes

This paper concerns dynamic optimisation for systems governed by Ito stochastic differential equations for which the pathwise uniqueness property holds. We study the relaxed control problem which is a generalization of the original problem where admissible controls are measure valued processes. In order for the relaxed problem to be truly an extension of the original one, the value functions for the two problems must be the same. For this purpose, we show under very general conditions on the coefficients that every relaxed diffusion is a strong limit of a sequence of diffusions associated with ordinary controls. As a consequence, it is proved that the value functions of both relaxed and original problems are equal.