Right inverses of L\'{e}vy processes

TL;DR

This paper characterizes when a right inverse exists for a Lévy process using its triplet, providing a clear criterion for the existence of partial right inverses.

Contribution

It establishes a necessary and sufficient condition for the existence of a partial right inverse of a Lévy process based on its Lévy triplet.

Findings

Provides a complete characterization of PRI existence

Links PRI existence to Lévy triplet parameters

Advances understanding of Lévy process path properties

Abstract

We call a right-continuous increasing process $K_x$ a partial right inverse (PRI) of a given L\'{e}vy process $X$ if $X_{K_x}=x$ for at least all $x$ in some random interval $[0,\zeta)$ of positive length. In this paper, we give a necessary and sufficient condition for the existence of a PRI in terms of the L\'{e}vy triplet.