Factorization representations in the boundary crossing problems for random walks on a Markov chain

Let tau be some stopping time for a random walk S-n defined on transitions of a finite Markov chain and let tau(t) be the first passage time across the level t which occurs after tau. We prove a theorem that establishes a connection between the dual Laplace-Stieltjes transforms of the joint distributions of (tau, S-tau) and (tau(t), S-tau(t)). This result applies to the study of the number of crossings of a strip by sample paths of a random walk.