Factorization Representations in the Boundary Crossing Problems for Random Walks on a Markov Chain

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