The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

Abstract

We study the asymptotic tail behaviour of the first passage time over a moving boundary for asymptotically \(\alpha \)-stable Levy processes with \(\alpha <1\). Our main result states that if the left tail of the Levy measure is regularly varying with index \(- \alpha \), and the moving boundary is equal to \(1 - t^{\gamma }\) for some \(\gamma <1/\alpha \), then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary \(1 + t^{\gamma }\) with \(\gamma <1/\alpha \) under the assumption of a regularly varying right tail with index \(-\alpha \).