First passage times of Levy processes over a one-sided moving boundary

#### F. Aurzada, T. Kramm, M. Savov

2015, v.21, Issue 1, 1-38

ABSTRACT

We study the tail of the distribution of the first passage time of a L\'evy process over a one-sided moving boundary.
Our main result states that if the boundary behaves as $t^{\gamma}$ for large $t$ for some $\gamma<1/2$ then the probability that the process stays below the boundary behaves asymptotically as in the case of a constant boundary. Both positive ($+t^\gamma$) and negative ($-t^\gamma$) boundaries are considered.

% To this aim, we develop a new technique using an iteration method to reduce the exponent $\gamma$ of the boundary in each step such that the boundary eventually turns into a constant boundary.

These results extend the findings of \cite{GreNov} and are motivated by results in the case of Brownian motion, for which the above result was proved in \cite{Uch}.

Keywords: boundary crossing probabilities; boundary crossing problem; first passage time; Levy processes; lower tail probabilities; moving boundary; one-sided boundary problem; one-sided exit problem; one-sided small deviations; survival exponent; persistence