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 A. Assuming time reversibility, we show that all the eigenvalues of A are real, with algebraic and geometric Multiplicities being the same, which allows LIS to identify the Jordan normal form of A. Furthermore, this fact simplifies the analysis of fluctuations of a MAP. We provide an illustrative example and show that our findings greatly reduce the computational efforts required to obtain A in the time-reversible case. (C) 2009 Elsevier B.V. All rights reserved.