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

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 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.