Реферат: We demonstrate an analytical approach to a number of problems related to crossing linear boundaries by the trajectory of a random walk. 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 joint distributions under study. The use of the Wiener-Hopf factorization is the main instrument to solve these identities. Thus we 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, continuity theorem 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 for the distributions under study. Some asymptotic results for the distributions of boundary functionals will be presented.

Библиографическая ссылка: Lotov V.I.
Operator method in boundary crossing problems for random walks
Международная научная конференция «Теория вероятностей, математическая статистика и приложения» 22-24 Apr 2024