Principle of smooth fit and diffusions with angles

Abstract

We show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problem fails to satisfy the smooth fit condition at the optimal stopping point b. On the other hand, if the scale function S of X is differentiable at b, then the smooth fit condition holds (whenever X is regular and G is differentiable at b). We give an example showing that the latter can happen even when at b.