Abstract
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.
Highlights
The Markov additive process can be seen as the Markov-modulated version of the Levy process
A common assumption in the vast literature on the extremes of spectrally onesided Markov additive processes (MAPs) is that the continuous-time Markov chain that serves as the background process is irreducible
When an independently evolving continuous-time Markov chain on d ∈ N states, usually referred to as the background process, is in state i, the MAP locally behaves as a Levy process Xi(·): a MAP allows for jumps at transition epochs of the background process
Summary
The Markov additive process (in this paper abbreviated to MAP) can be seen as the Markov-modulated version of the Levy process. In the context of credit risk, one could think of companies paying interest to an obligor until they go into default, causing a loss to the obligor, after which they effectively leave the system—another setting that can be modeled using a nonirreducible background process This credit-related example motivated Delsing and Mandjes (2021) to consider the extreme values attained by a MAP of a specific Cramer–Lundberg type endowed with a nonirreducible background process of a specific structure. The second key result, as established by Ivanovs et al (2010), characterizes the number of singularities with positive real parts of the matrix exponent corresponding to a spectrally one-sided MAP Using this result, we find a procedure of obtaining the solution to the system of equations, determining the distribution of Zi for all i.
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