Stochastic control problems with unbounded control set

Stochastic differential equations with adapted control are obtained as natural limits of birth and death processes, and an important class of such SDEs in which the control set is unbounded is studied. A change of time technique allows us to transform such control problems to ones with a bounded control set. The transformed problem has the same value function as the originally posed problem. A nice feature of the transformed problem consists in (a) the existence of an optimal control that is expressible in simple terms, and (b) the existence of a natural sequence of physically realizable controls for which the payoff functions converge to the value function.