On the convergence from discrete to continuous time in an optimal stopping problem

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 Vh(x), respectively. We identify the rate of convergence of Vh(x) to V(x) and the rate of convergence of the stopping regions, and provide simple formulas for the rate coefficients.