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A Review on Phase-type Distributions and their Use in Risk Theory

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  • Bladt, Mogens

Abstract

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

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  • Bladt, Mogens, 2005. "A Review on Phase-type Distributions and their Use in Risk Theory," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 145-161, May.
    • Handle: RePEc:cup:astinb:v:35:y:2005:i:01:p:145-161_01
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    Cited by:

    1. Weng, Chengguo, 2013. "Constant proportion portfolio insurance under a regime switching exponential Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 508-521.
    2. Jukka Lempa, 2020. "Some results on optimal stopping under phase-type distributed implementation delay," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(3), pages 559-583, June.
    3. Eric C. K. Cheung & Oscar Peralta & Jae-Kyung Woo, 2021. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Papers 2201.11122, arXiv.org.
    4. Kim, Joseph H.T. & Kim, Joocheol, 2015. "A parametric alternative to the Hill estimator for heavy-tailed distributions," Journal of Banking & Finance, Elsevier, vol. 54(C), pages 60-71.
    5. Anna Castañer & M.Mercè Claramunt & Maite Mármol, 2014. "Some optimization and decision problems in proportional reinsurance," UB School of Economics Working Papers 2014/310, University of Barcelona School of Economics.
    6. Gardner, Clara Brimnes & Nielsen, Sara Dorthea & Eltved, Morten & Rasmussen, Thomas Kjær & Nielsen, Otto Anker & Nielsen, Bo Friis, 2021. "Calculating conditional passenger travel time distributions in mixed schedule- and frequency-based public transport networks using Markov chains," Transportation Research Part B: Methodological, Elsevier, vol. 152(C), pages 1-17.
    7. Brigo, Damiano & Mai, Jan-Frederik & Scherer, Matthias, 2016. "Markov multi-variate survival indicators for default simulation as a new characterization of the Marshall–Olkin law," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 60-66.
    8. Marios N. Kyriacou, 2015. "Credit Risk Measurement in Financial Institutions: Going Beyond Regulatory Compliance," Cyprus Economic Policy Review, University of Cyprus, Economics Research Centre, vol. 9(1), pages 31-72, June.
    9. Ahn, Soohan & Kim, Joseph H.T. & Ramaswami, Vaidyanathan, 2012. "A new class of models for heavy tailed distributions in finance and insurance risk," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 43-52.
    10. Wanlu Gu & Neng Fan & Haitao Liao, 2019. "Evaluating readmission rates and discharge planning by analyzing the length-of-stay of patients," Annals of Operations Research, Springer, vol. 276(1), pages 89-108, May.
    11. Paul Embrechts & Marco Frei, 2009. "Panjer recursion versus FFT for compound distributions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(3), pages 497-508, July.
    12. Pavel V. Shevchenko, 2010. "Calculation of aggregate loss distributions," Papers 1008.1108, arXiv.org.
    13. Franck Adékambi & Kokou Essiomle, 2020. "Ruin Probability for Stochastic Flows of Financial Contract under Phase-Type Distribution," Risks, MDPI, vol. 8(2), pages 1-21, May.
    14. Søren Asmussen & Patrick J. Laub & Hailiang Yang, 2019. "Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits," Risks, MDPI, vol. 7(1), pages 1-22, February.
    15. Cheung, Eric C.K. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.

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