On optimal threshold stopping times for Ito diffusions

The paper deals with an optimal stopping problem for one-dimensional Ito diffusion process and terminal payoff function. We study the following problem: under what conditions the stopping time which is optimal over the class of threshold strategies (specifying by first time when the underlying process exceeds some level) remains optimal over all stopping times. We prove that excessiveness of payoff function is the only necessary and sufficient condition which connects optimality over the class of threshold stopping times and over all stopping times.