Stochastic barriers for the Wiener process and a mathematical model

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<Superscript>p0≤t≤TW(t)−[f(t)+X(t)]≧0</Superscript> when X(t) is a stochastic process. By taking a compound Poisson process as X(t), an interesting mathematical model is created.