On L\'evy processes conditioned to avoid zero

TL;DR

This paper constructs and analyzes the law of a Lévy process conditioned to avoid zero, using two methods, and explores properties and examples such as stable and spectrally negative Lévy processes.

Contribution

It introduces two new constructions for Lévy processes conditioned to avoid zero, extending previous symmetric case results and providing detailed properties and examples.

Findings

The conditioned process exists under mild conditions.

The $h$-transformation approach generalizes previous results.

Explicit examples include stable and spectrally negative Lévy processes.

Abstract

The purpose of this paper is to construct the law of a L\'evy process conditioned to avoid zero, under mild technicals conditions, two of them being that the point zero is regular for itself and the L\'evy process is not a compound Poisson process. Two constructions are proposed, the first lies on the method of $h$-transformation, which requires a deep study of the associated excessive function; while in the second it is obtained by conditioning the underlying L\'evy process to avoid zero up to an independent exponential time whose parameter tends to $0.$ The former approach generalizes some of the results obtained by Yano \cite{Yano10} in the symmetric case and recovers some of main results in Yano's work \cite{Yano13}, while the latter is reminiscent of \cite{Chaumont-Doney05}. We give some properties of the resulting process and we describe in some detail two examples: alpha stable and spectrally negative L\'evy processes.