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We study the exact asymptotics for the distribution of the first time$\tau_x$ a L\'evy 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 acertain function $V(x)$. Using known results for the large deviations of randomwalks we obtain asymptotics for $\mathbf P(\tau_x>t)$ explicitly in both lightand heavy tailed cases. We also apply our results to find asymptotics for thedistribution of the busy period in an M/G/1 queue.

Asymptotics for first-passage times of L\'evy processes and random walks