We consider the problem of optimal stopping for a one-dimensional diffusion process. Two classes of admissible stopping times are considered. The first class consists of all nonanticipating stopping times that take values in [0, ∞], while the second class further restricts the set of allowed values to the discrete grid {nh:n = 0,1,2,...,∞} for some parameter h> 0. The value functions for the two problems are denoted by V (x) and V h (x), respectively. We identify the rate of convergence of V h (x) to V (x) and the rate of convergence of the stopping regions, and provide simple formulas for the rate coefficients. 1. Introduction. One

This paper studies optimal stopping problems for general diffusion processes with an uncertain time horizon and with different levels of information based on which the decision to stop or to continue is made. First, corresponding value functions are com-pared and related explicitly to their counterparts without the time uncertainty. Second, to analyze optimal stopping strategies, characterization results regarding connectivity of the “stopping ” and “continuation ” regions are derived. In particular, sufficient con-ditions for the well-known “threshold type ” optimal stopping rule are given, based on which explicit solutions are derived for problems with standard American call/put type payoff functions.