Alternative approach to derive q-potential measures of refracted spectrally L\'evy processes

TL;DR

This paper introduces a novel method to derive q-potential measures for refracted spectrally Lévy processes that does not rely on scale functions, enabling broader applicability to general refracted Lévy processes.

Contribution

It presents a new approach to obtain q-potential measures without using scale functions, extending the analysis to more general refracted Lévy processes.

Findings

Derivation of q-potential measures without scale functions

Extension to general refracted Lévy processes

Applicable to spectrally negative Lévy processes

Abstract

For a refracted L\'evy process driven by a spectrally negative L\'evy process, we use a different approach to derive expressions for its q-potential measures without killing. Unlike previous methods whose derivations depend on scale functions which are defined only for spectrally negative L\'evy processes, our approach is free of scale functions. This makes it possible to extend the result here to a quite general refracted L\'evy process by applying the approach presented below.