Extreme Value Analysis for a Markov Additive Process Driven by a Nonirreducible Background Chain

A common assumption in the vast literature on the extremes of spectrally one-sided Markov additive processes (MAPs) is that the continuous-time Markov chain that serves as the background process is irreducible. In the present paper, we consider, motivated by, for example, applications in credit risk, the case in which the irreducibility condition has been lifted, thus allowing the presence of one or more transient classes. More specifically, we consider the distribution of the maximum when the MAP under study has only positive jumps (the spectrally positive case) or negative jumps (the spectrally negative case). The methodology used relies on two crucial previous results: (i) the Wiener–Hopf decomposition for Lévy processes and, in particular, its explicit form in spectrally one-sided cases and (ii) a result on the number of singularities of the matrix exponent of a spectrally one-sided MAP. In both the spectrally positive and negative cases, we derive a system of linear equations of which the solution characterizes the distribution of the maximum of the process. As a by-product of our results, we develop a procedure for calculating the maximum of a spectrally one-sided Lévy process over a phase-type distributed time interval.