Approximation and duality problems of refracted processes

TL;DR

This paper introduces a generalized class of refracted processes derived from two standard processes without positive jumps, explores their approximation via jump removal, and examines conditions for their duality.

Contribution

It constructs a new class of refracted processes using excursion theory and analyzes their approximation and duality properties.

Findings

Refracted processes generalize Kyprianou--Loeffen's models.

Approximation by removing small jumps converges to the original process.

Conditions for the existence of dual processes are established.

Abstract

For given two standard processes with no positive jumps, we construct, using the excursion theory, a Markov process whose positive and negative motions have the same law as the two processes. The resulting process is a generalization of Kyprianou--Loeffen's refracted L\'evy processes. We discuss approximation problem for our refracted processes coming from L\'evy processes by removing small jumps and taking the limit as the removal level tends to zero. We also discuss conditions for refracted processes to have dual processes.