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.