Asymptotic expansions for the distribution of the crossing number of a strip by a Markov-modulated random walk

We obtain complete asymptotic expansions for the distribution of the crossing number of a strip in $n$ steps by sample paths of a random walk defined on a finite Markov chain. We assume that the Cramer condition holds for the distribution of jumps and the width of the strip grows with $n$. The method consists in finding factorization representations of the moment generating functions of the distributions under study, isolating the main terms in the asymptotics of the representations, and inverting those main terms by the modified saddle-point method.