Diffusion approximations for the maximum of a perturbed random walk

Abstract

Consider a random walk S=(S n : n≥0) that is ‘perturbed’ by a stationary sequence (ξ n : n≥0) to produce the process S=(S n +ξ n : n≥0). In this paper, we are concerned with developing limit theorems and approximations for the distribution of M n =max{S k +ξ k : 0≤k≤n} when the random walk has a drift close to 0. Such maxima are of interest in several modeling contexts, including operations management and insurance risk theory. The associated limits combine features of both conventional diffusion approximations for random walks and extreme-value limit theory.