Asymptotics for the First Passage Times of Lévy Processes and Random Walks

Abstract

We study the exact asymptotics for the distribution of the first time, τ x , a Lévy process X t crosses a fixed negative level -x. We prove that ℙ{τ x >t} ~V(x) ℙ{X t ≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τ x >t} explicitly in both light- and heavy-tailed cases.