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}$forsome$\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 $\gamma <1\mathrm{/}\alpha$ under the assumption of a regularly varying right tail with index $-\alpha$.