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 Lévy processes with $\alpha <1$. Our main result states that if the left tail of the Lévy 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 $-\alpha$.

correction of earlier mistakes, in particular the result can only be claimed for α<1, to appear in: Journal of Theoretical Probability

References in corpus (5)

• Persistence probabilities \& exponents
• First passage times of Lévy processes over a one-sided moving boundary
• Exact asymptotics for the instant of crossing a curve boundary by an asymptotically stable random walk
• Asymptotics for first-passage times of Lévy processes and random walks
• Small and Large Time Stability of the Time taken for a Lévy Process to Cross Curved Boundaries

Cited by in corpus (1)

• Exact asymptotics for the instant of crossing a curve boundary by an asymptotically stable random walk