Generalized refracted L\'evy process and its application to exit problem

TL;DR

This paper introduces a new class of refracted Lévy processes with distinct positive and negative motions, analyzes their exit problems, potential measures, and discusses approximation methods, extending previous models in stochastic process theory.

Contribution

It generalizes existing refracted Lévy processes by allowing different Lévy processes for positive and negative motions, using excursion theory for construction.

Findings

Derived potential measures for the new process

Solved the exit problem for the generalized process

Discussed approximation techniques

Abstract

Generalizing Kyprianou--Loeffen's refracted L\'evy processes, we define a new refracted L\'evy process which is a Markov process whose positive and negative motions are L\'evy processes different from each other. To construct it we utilize the excursion theory. We study its exit problem and the potential measures of the killed processes. We also discuss approximation problem.