Proper two-sided exits of a L\'evy process

TL;DR

This paper characterizes when two-sided exit times of a Lévy process are proper and identifies conditions under which the supports of these exit times are unbounded and contain zero.

Contribution

It provides a complete characterization of proper two-sided exits of Lévy processes and describes conditions for unbounded supports of first exit times from bounded annuli.

Findings

Two-sided exits are proper iff the process is not a subordinator or its negative.

Supports of first exit times can be unbounded and include zero under certain conditions.

Characterization of Lévy processes with specific support properties for exit times.

Abstract

It is proved that the two-sided exits of a Levy process are proper, i.e. not a.s. equal to their one-sided counterparts, if and only if said process is not a subordinator or the negative of a subordinator. Furthermore, Levy processes are characterized, for which the supports of the first exit times from bounded annuli, simultaneously on each of the two events corresponding to exit at the lower and the upper boundary, respectively are unbounded, contain $0$, are equal to $[0,\infty)$.