If the problem of optimal stopping for finite state discrete time Markov Chains is solved on the infinite horizon explicitly?
Edited: This means if for a given MC $X(n)$ with a state space $x_1,...,x_n$, transition matrix $P$; and a vector $g_1 = g(x_1),...,g_n = g(X_n)$ there is an explicit solution to the problem $ v_i = \sup\limits_{\tau<\infty}\mathsf{E}[g(X(\tau))|X_0 = x_i]. $