The excursion measure away from zero for spectrally negative L\'evy processes

TL;DR

This paper characterizes the excursion measure from zero for spectrally negative Lévy processes by relating it to conditioned processes and reflected processes, providing a comprehensive description of their excursion behavior.

Contribution

It introduces a novel description of the excursion measure from zero for spectrally negative Lévy processes using conditioned and reflected processes, linking various excursion measures.

Findings

Connected excursion measures from zero with reflected processes.

Described the entrance law at zero for conditioned processes.

Established relationships between excursion measures and reflected processes.

Abstract

We provide a description of the excursion measure from a point for a spectrally negative L\'evy process. The description is based in two main ingredients. The first is building a spectrally negative L\'evy process conditioned to avoid zero and the study of its entrance law at zero. The latter is connected with both the excursion measure from zero of the process reflected in its infimum and reflected in its supremum. This lead us to establish a connection with the excursion measure from the state zero with the excursion measure from zero for the process reflected at the infimum and reflected at the supremum, respectively, which is the second main ingredient of our description.