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Lévy systems and the time value of ruin for Markov additive processes

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Abstract

In this paper we study the ruin problem for an insurance risk process driven by a spectrally-positive Markov additive process. Particular attention is given to the family of spectrally-positive Markov-modulated Lévy processes. We give an expression for the expected discounted penalty function by extending results available in the literature. In particular, we generalize some results in Biffis and Kyprianou (Insur Math Econ 46:85–91, 2010) to a more general setting provided by the theory of Markov additive processes. This natural extension is possible thanks to the concept of Lévy systems that allows us to generalize well-known results for Lévy processes to a larger family of Markov additive processes. We also discuss how more compact expressions for the expected discounted penalty function can be obtained using the notion of scale matrix of a Markov additive process.

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Acknowledgments

The authors would like to thank the anonymous reviewers for a careful reading of the manuscript and for their comments and suggestions that helped us improve the paper substantially. This research was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) operating Grant RGPIN-311660 and by the Mathematical Sciences Network of Excellence MPRIME. Zied Ben Salah acknowledges financial support from the Tunisian government via the Ph.D. fellowship program.

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  1. Department of Mathematics and Statistics, University of Montreal, CP. 6128 succ. centre-ville, Montreal, QC, H3C 3J7, Canada

    Zied Ben Salah & Manuel Morales

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  1. Zied Ben Salah
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  2. Manuel Morales
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Correspondence to Manuel Morales.

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Ben Salah, Z., Morales, M. Lévy systems and the time value of ruin for Markov additive processes. Eur. Actuar. J. 2, 289–317 (2012). https://doi.org/10.1007/s13385-012-0053-5

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  • DOI: https://doi.org/10.1007/s13385-012-0053-5

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