Optimality of Threshold Stopping Times for Diffusion Processes

This paper is concerned with the optimal stopping problem for Itô diffusion processes over a class of stopping times. Necessary and sufficient optimality conditions are studied for a parametrically specified class of stopping times. A detailed analysis is given for the case of one-dimensional diffusion processes and threshold stopping times. Necessary and sufficient conditions are put forward for optimality of a threshold stopping time over all stopping times. A number of relations are obtained between the solution of the optimal stopping problem over the class of threshold moments and the solution of the free-boundary problem.