First passage of time-reversible spectrally negative Markov additive processes

Authors:
Jevgenijs Ivanovs;Michel Mandjes
Affiliations:
Eurandom, Eindhoven University of Technology, Netherlands and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Netherlands;Eurandom, Eindhoven University of Technology, Netherlands and Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Netherlands
Venue:
Operations Research Letters
Year:
2010

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Abstract

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 @L. Assuming time reversibility, we show that all the eigenvalues of @L are real, with algebraic and geometric multiplicities being the same, which allows us to identify the Jordan normal form of @L. 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 @L in the time-reversible case.