Weakly coercive problems in nonlinear stochastic control: Existence of optimal controls.

In this paper we prove the existence of an optimal control for some nonlinear stochastic control problems where the control set is unbounded, but instead of a classical coercivity hypothesis, weaker assumptions are made. Our model includes singular stochastic optimization control problems, such as, for instance, the so-called monotone follower problem or the additive control problem, extensively treated in the literature, mostly in the case of linear systems. We apply a technique of time transformation which allows us to show that the original problems are equivalent to some optimal stopping time problems with bounded controls.