Stability of overshoots of Markov additive processes

TL;DR

This paper establishes stability results for overshoots of Markov additive processes, extending fluctuation theory and Wiener-Hopf factorization, with applications to self-similar Markov processes and stable process sampling.

Contribution

It extends the fluctuation theory of MAPs, generalizes Wiener-Hopf factorization, and applies these results to self-similar Markov processes and stable process analysis.

Findings

Proved stability of overshoots for MAPs with finite modulating space.

Extended Wiener-Hopf factorization for MAPs.

Analyzed mixing behavior of stable processes at hitting times.

Abstract

We prove precise stability results for overshoots of Markov additive processes (MAPs) with finite modulating space. Our approach is based on the Markovian nature of overshoots of MAPs whose mixing and ergodic properties are investigated in terms of the characteristics of the MAP. On our way we extend fluctuation theory of MAPs, contributing among others to the understanding of the Wiener-Hopf factorization for MAPs by generalizing Vigon's \'equations amicales invers\'es known for L\'evy processes. Using the Lamperti transformation the results can be applied to self-similar Markov processes. Among many possible applications, we study the mixing behavior of stable processes sampled at first hitting times as a concrete example.