Sample-path large deviations for L\'evy processes and random walks with Weibull increments

TL;DR

This paper establishes advanced large deviations principles for Lévy processes and random walks with Weibull-type heavy-tailed jumps, providing new topological results and applications to first passage problems.

Contribution

It extends large deviations principles to Weibull-tailed Lévy processes in multiple topologies and introduces a quasi-variational rate function representation.

Findings

Extended LDP in J1 topology

Full LDP in M1' topology

Application to first passage problem

Abstract

We study sample-path large deviations for L\'evy processes and random walks with heavy-tailed jump-size distributions that are of Weibull type. Our main results include an extended form of an LDP (large deviations principle) in the $J_1$ topology, and a full LDP in the $M_1'$ topology. The rate function can be represented as the solution to a quasi-variational problem. The sharpness and applicability of these results are illustrated by a counterexample proving the nonexistence of a full LDP in the $J_1$ topology, and by an application to a first passage problem.