First passage of time-reversible spectrally-negative Markov additive processes

We study the first passage process of a spectrally-negative Markov additive process (MAP). The focus is on the background Markov chain at the times of the first passage. This process is a Markov chain itself with a transition rate matrix ¿. Assuming time-reversibility we show that all the eigenvalues of ¿ are real with algebraic and geometric multiplicities being the same, which allows us to identify the Jordan normal form of ¿. Furthermore, this fact simplifies the analysis of uctuations of a MAP. We provide an illustrative example and show that our findings greatly reduce the computational efforts required to obtain ¿ in the time-reversible case.