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On scale functions for Lévy processes with negative phase-type jumps

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Abstract

We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

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Acknowledgements

The financial support of Sapere Aude Starting Grant 8049-00021B “Distributional Robustness in Assessment of Extreme Risk” from Independent Research Fund Denmark is gratefully acknowledged.

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  1. Department of Mathematics, Aarhus University, Aarhus, Denmark

    Jevgenijs Ivanovs

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  1. Jevgenijs Ivanovs
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Correspondence to Jevgenijs Ivanovs.

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Ivanovs, J. On scale functions for Lévy processes with negative phase-type jumps. Queueing Syst 98, 3–19 (2021). https://doi.org/10.1007/s11134-021-09696-w

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