Sample path large deviations for Lévy processes and random walks with Weibull increments

We study sample path large deviations for Lévy processes and random walks with heavy-tailed jump-size distributions that are of Weibull type. The sharpness and applicability of these results are illustrated by a counterexample proving the nonexistence of a full LDP in the J1 J 1 topology, and by an application to a first passage problem