The first passage time problem over a moving boundary for asymptotically stable L\'evy processes

TL;DR

This paper investigates the asymptotic behavior of the first-passage time over moving boundaries for certain stable Lévy processes, revealing that the tail behavior remains consistent with constant boundaries under specific conditions.

Contribution

It provides new asymptotic results for first-passage times over moving boundaries for asymptotically stable Lévy processes with tail conditions.

Findings

Probability decay rate matches constant boundary case for specific moving boundaries.

Results apply to both increasing and decreasing boundaries with certain regular variation conditions.

Extends understanding of boundary crossing probabilities for stable Lévy processes.

Abstract

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically $\alpha$-stable L\'evy processes with $\alpha<1$. Our main result states that if the left tail of the L\'evy 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 $- \alpha$.