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First passage times for Markov additive processes with positive jumps of phase type. (English) Zbl 1156.60059
The author extends some results holding for spectrally negative Lévy processes to Markov Additive Processes (MAPs). An important part is played by the first passage times, which are determined in terms of their Laplace transform. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration for computing this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.

MSC:
60J25 Continuous-time Markov processes on general state spaces
60J55 Local time and additive functionals
60G51 Processes with independent increments; Lévy processes
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