Splitting and time reversal for Markov additive processes

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 the behaviour of the process, and while being structurally similar to the Lévy 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. Restricting to the case of no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line. The latter result is obtained using three quite different approaches based on the established path decomposition theory, which further demonstrates its applicability.