A Factorization Method in Boundary Crossing Problems for Random Walks

We demonstrate an analytical approach to a number of problems related to crossing linear boundaries by trajectories of a random walk. The main results consist in finding explicit expressions and asymptotic expansions for distributions of various boundary functionals such as first exit time and overshoot, the crossing number of a strip, sojourn time, etc. The method includes several steps. We start with the identities containing Laplace transforms of the joint distributions under study. Wiener-Hopf factorization is the main tool for solving these identities. We thus obtain explicit expressions for the Laplace transforms in terms of factorization components. It turns out that in many cases Laplace transforms are expressed through the special factorization operators which are of particular interest. We further discuss possibilities of exact expressions for these operators, analyze their analytic structure, and obtain asymptotic representations for them under the assumption that the boundaries tend to infinity. After that we invert Laplace transforms asymptotically to get limit theorems and asymptotic expansions, including complete asymptotic expansions.