A stochastic control problem with different value functions for singular and absolutely continuous control

A stochastic control problem is obtained as a "small noise" approximation to a deterministic optimal control problem. Two classes of admissible controls are considered and the optimal control policies are explicitly determined for each admissible class. The larger admissible class contains controls referred to as singular stochastic controls. For this class, the cumulative effect of control has bounded variation trajectories. The smaller admissible class contains the standard stochastic controls whose cumulative effect has absolutely continuous trajectories. These controls are referred to as absolutely continuous controls. The optimal singular control provides a cost strictly smaller than the minimum cost achievable when only absolutely continuous stochastic controls are admissible. In particular, this shows that it is not always possible to approach the optimal cost for singular control if only the standard stochastic control policies are admissible.