On the new approach to the optimal stopping problem

We consider the optimal stopping problem of a Markov chain with an infinite horizon and the payoff function g(z). Sonin I.M. for the case of finit number m of states proposed an algorithm which allows one to find the value function and the stopping set in no more than 2m steps. This algorithm on each step considers a new chain with a new state space, eliminating the states where we know we should not stop. We discuss the case of an arbitrary state space, which requires a modification of the algorithm.
Consider, as Sonin did, set C_1 where the reward of an immediate stopping is less than the reward of stopping at the first step. At this set we know we should not stop. Let g_1(z) be the reward of stopping at the moment of the first visit to the set complimentary for C_1. Then under certain assumptions g_1(z) is greater than g(z) for z from C_1 and is equal to g(z) on the complimentary set. For the same chain consider the problem with a new payoff function which is equal to g_1(z). Obviously this problem is equivalent to the initial one. Let us now add to the set C_1 the points where the reward of an immediate stopping is less than the reward of stopping at the first step with payoff function g_1(z). We get set C_2 and corresponding function g_2(z). Finally we get sequence of functions g_n(z) and sets C_n which are nondecreasing and converge correspondingly to the value function and the continuation set of the initial problem.
We show the efficiency of this procedure and possibilities for generalizations in continuous time.