  This paper is concerned with the solution of the optimal stopping problem
associated to the valuation of Perpetual American options driven by continuous
time Markov chains. We introduce a new dynamic approach for the numerical
pricing of this type of American options where the main idea is to build a
monotone sequence of almost excessive functions that are associated to hitting
times of explicit sets. Under minimal assumptions about the payoff and the
Markov chain, we prove that the value function of an American option is
characterized by the limit of this monotone sequence.
