Asymptotics for first-passage times of Lévy processes and random walks

We study the exact asymptotics for the distribution of the first time, tau_x, a Levy process X_t crosses a fixed negative level -x. We prove that P{tau_x >t\} \sim V(x) P\{X_t>0\}/t as t\to\infty for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for P{tau_x>t} explicitly in both light- and heavy-tailed cases.