On an approach to boundary crossing by stochastic processes

Abstract

In this paper we provide an overview as well as new (definitive) results of an approach to boundary crossing. The first published results in this direction appeared in de la Peña and Giné (1999) book on decoupling. They include order of magnitude bounds for the first hitting time of the norm of continuous Banach-Space valued processes with independent increments. One of our main results is a sharp lower bound for the first hitting time of càdlàg real-valued processes X(t), where X(0)=0 with arbitrary dependence structure: ETrγ≥∫01{a−1(rα)}γdα, where Tr=inf{t>0:X(t)≥r},a(t)=E{sup0≤s≤tX(s)} and γ>0. Under certain extra conditions, we also obtain an upper bound for ETrγ. As the main text suggests, although Tr is defined as the hitting time of X(t) hitting a level boundary, the bounds developed can be extended to more general processes and boundaries. We shall illustrate applications of the bounds derived for additive processes, Gaussian Processes, Bessel Processes, Bessel bridges among others. By considering the non-random function a(t), we can show that in various situations, ETr≈a−1(r).