On the optimal stopping problem for one-dimensional diffusions

A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessive functions are essentially concave functions in some suitable generalized sense, and vice–versa. This, in turn, permits a characterization of the value function of the optimal stopping problem as “the smallest nonnegative concave majorant of the reward function”, and allows us to generalize results of Dynkin–Yushkevich for the standard Brownian motion. Moreover, we show how to reduce the discounted optimal stopping problems for an arbitrary diffusion process, to an undiscounted optimal stopping problem for the standard Brownian motion. The concavity of the value functions also leads to conclusions about their smooth-ness, thanks to the properties of concave functions. One is thus led to a new per-spective and new facts about the smooth–fit principle in the context of optimal stopping. The results are illustrated in detail on a number of non–trivial, concrete optimal stopping problems, both old and new