Stochastic barriers for the Wiener process

Abstract

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ TW(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ TW(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.