Passage time and fluctuation calculations for subexponential L\'evy processes

TL;DR

This paper analyzes passage times for heavy-tailed Lévy processes, deriving limit distributions and connecting results to extreme value theory, under mild assumptions including drift to negative infinity and subexponential growth.

Contribution

It provides new limit theorems for passage times and overshoots of Lévy processes with heavy tails, linking these to extreme value theory and regular variation.

Findings

Limit distributions for passage times and overshoots are established.

Results hold under mild conditions including subexponential tail growth.

Connections to extreme value theory are demonstrated.

Abstract

We consider the passage time problem for L\'evy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth on the positive side. Local and functional versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level. A significant connection is made with extreme value theory via regular variation or maximum domain of attraction conditions imposed on the positive tail of the canonical measure, which are shown to be necessary for the kind of convergence behaviour we are interested in.