A Factorization Method in Boundary Crossing Problems for Random Walks

V.I. Lotov

2019, v.25, Issue 4, 709-722

ABSTRACT

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\tire 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.

Keywords: random walk, boundary crossing problems, factorization method, asymptotic expansions

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