Exact formulas in some boundary crossing problems for integer-valued random walks

For a wide class of integer-valued random walks, we obtain exact expressions for the distribution of the first excess over level and the corresponding renewal function as well as for the distribution of the trajectory supremum if it is finite. We discuss possibilities of obtaining explicit expressions for pre-stationary and stationary distributions of a random walk with switchings at the strip boundaries. The research is based on the factorization representations for the double moment generating functions of the distributions under study.