A note on refracted L\'evy processes without positive jumps

TL;DR

This paper derives new formulas for the q-potential measures of refracted spectrally negative Levy processes that do not rely on q-scale functions, potentially applicable to more general Levy processes.

Contribution

It introduces formulas for q-potential measures without using q-scale functions, expanding applicability beyond previous methods.

Findings

Formulas are free of q-scale functions.

Results potentially extend to general refracted Levy processes.

Enhances analytical tools for Levy process analysis.

Abstract

For a refracted spectrally negative Levy process, we find some new and fantastic formulas for its q-potential measures without killing. Unlike previous results, which are written in terms of the known q-scale functions, our formulas are free of the q-scale functions. This makes our results become extremely important since it is likely that our formulas also hold for a general refracted Levy process.