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

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 $- \alpha$.