The first passage time problem over a moving boundary for asymptotically stable Lévy processes

We study the asymptotic tail behaviour of the first-passage time over a moving boundary for asymptotically $α$-stable Lévy processes with $α<1$. Our main result states that if the left tail of the Lévy measure is regularly varying with index $- α$ and the moving boundary is equal to $1 - t^γ$ for some $γ<1/α$, 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 $1 + t^γ$ with $γ<1/α$ under the assumption of a regularly varying right tail with index $- α$.

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