# On weighted occupation times for refracted spectrally negative L\'evy processes

## TL;DR

This paper derives explicit formulas for weighted occupation times of refracted spectrally negative Lévy processes, expressing key quantities via integral equations involving scale functions, thereby extending previous results.

## Contribution

It introduces a novel approach to compute weighted occupation times for refracted spectrally negative Lévy processes using integral equations and scale functions.

## Key findings

- Explicit Laplace transform expressions derived
- Integral equations involving weight and scale functions established
- Previous results on refracted Lévy processes recovered

## Abstract

For refracted spectrally negative L\'evy processes, we identify expressions of several quantities related to Laplace transforms on their weighted occupation times until first exit times. Such quantities are expressed in terms of unique solutions to integral equations involving weight functions and scale functions for the associated spectrally negative L\'evy processes. Previous results on refracted L\'evy processes are recovered.