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

Abstract

We study the exact asymptotics for the distribution of the first time ${\tau }_x$ a Lévy process $X_t$ crosses a negative level $-x$. We prove that $\mathbf{P}({\tau }_x>t)\sim V(x)\mathbf{P}(X_t\ge 0)/t$ as $t\to \infty$ for a certain function $V(x)$. Using known results for the large deviations of random walks we obtain asymptotics for $\mathbf{P}({\tau }_x>t)$ explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue.