A technique for stochastic control problems with unbounded control set

We describe a change of time technique for stochastic control problems with unbounded control set. We demonstrate the technique on a class of maximiza tion problems that do not have optimal controls. Given such a problem, we i ntroduce an extended problem which has the same value function as the origi nal problem and for which there exist optimal controls that are expressible in simple terms. This device yields a natural sequence of suboptimal contr ols for the original problem. By this we mean a sequence of controls for wh ich the payoff functions approach the value function.