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.