Exact tail asymptotics of the supremum attained by a Lévy process

In this short communication we analyze the tail asymptotics corresponding to the maximum value attained by a Lévy process with negative drift. The note has two contributions: a short and elementary proof of these asymptotics, and an importance sampling algorithm to estimate the rare-event probabilities under consideration. Keywords: Lévy processes; Second factorization identity; Tail asymptotics; Importance samplin