Splitting and time reversal for Markov additive processes

TL;DR

This paper investigates the path structure of Markov additive processes with finite phases, establishing splitting properties at extrema and passage times, and deriving phase-dependent Wiener-Hopf factorizations with applications to last exit times.

Contribution

It introduces new path decomposition results for Markov additive processes, extending classical Lévy process techniques to phase-dependent settings with explicit formulas.

Findings

Established splitting laws at extrema and passage times conditioned on phase.

Derived phase-dependent Wiener-Hopf factorization formulas.

Characterized last exit times from negative half-line using multiple methods.

Abstract

We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into behavior of the process, and while being structurally similar to the L\'evy process case, they demonstrate various new features. As an application we formulate the Wiener-Hopf factorization, where time is counted in each phase separately and killing of the process is phase dependent. Assuming no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line using three different approaches, which further demonstrates applicability of path decomposition theory.