Markov additive friendships

TL;DR

This paper investigates the inverse Wiener--Hopf factorisation problem for Markov additive processes, providing a complete solution, simpler construction conditions, and insights into the uniqueness of the factorisation.

Contribution

It extends Vigon's theory of friendship from Lévy processes to Markov additive processes, offering a comprehensive answer to the inverse problem and conditions for process construction.

Findings

Complete solution to the inverse Wiener--Hopf problem for MAPs

Simpler sufficient conditions for process construction via friendship

Partial results on the uniqueness of Wiener--Hopf factorisation

Abstract

The Wiener--Hopf factorisation of a L\'evy or Markov additive process describes the way that it attains new maxima and minima in terms of a pair of so-called ladder height processes. Vigon's theory of friendship for L\'evy processes addresses the inverse problem: when does a process exist which has certain prescribed ladder height processes? We give a complete answer to this problem for Markov additive processes, provide simpler sufficient conditions for constructing processes using friendship, and address in part the question of the uniqueness of the Wiener--Hopf factorisation for Markov additive processes.