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Phase-type Approximations to Finite-time Ruin Probabilities in the Sparre-Andersen and Stationary Renewal Risk Models

Author

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  • Stanford, D.A.
  • Avram, F.
  • Badescu, A.L.
  • Breuer, L.
  • Silva Soares, A. Da
  • Latouche, G.

Abstract

The present paper extends the “Erlangization†idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

Suggested Citation

  • Stanford, D.A. & Avram, F. & Badescu, A.L. & Breuer, L. & Silva Soares, A. Da & Latouche, G., 2005. "Phase-type Approximations to Finite-time Ruin Probabilities in the Sparre-Andersen and Stationary Renewal Risk Models," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 131-144, May.
    • Handle: RePEc:cup:astinb:v:35:y:2005:i:01:p:131-144_01
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    Cited by:

    1. Mehmet Akif Yazici & Nail Akar, 2017. "The finite/infinite horizon ruin problem with multi-threshold premiums: a Markov fluid queue approach," Annals of Operations Research, Springer, vol. 252(1), pages 85-99, May.
    2. Zhimin Zhang & Eric C. K. Cheung, 2016. "The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 275-306, June.
    3. Avanzi, Benjamin & Cheung, Eric C.K. & Wong, Bernard & Woo, Jae-Kyung, 2013. "On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 98-113.
    4. Bladt, Mogens & Ivanovs, Jevgenijs, 2021. "Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 105-123.
    5. Choi, Michael C.H. & Cheung, Eric C.K., 2014. "On the expected discounted dividends in the Cramér–Lundberg risk model with more frequent ruin monitoring than dividend decisions," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 121-132.
    6. Hansjörg Albrecher & José Carlos Araujo-Acuna, 2022. "On The Randomized Schmitter Problem," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 515-535, June.
    7. Cheung, Eric C.K. & Wong, Jeff T.Y., 2017. "On the dual risk model with Parisian implementation delays in dividend payments," European Journal of Operational Research, Elsevier, vol. 257(1), pages 159-173.
    8. Yaodi Yong & Hailiang Yang, 2021. "Valuation of Cliquet-Style Guarantees with Death Benefits in Jump Diffusion Models," Mathematics, MDPI, vol. 9(16), pages 1-21, August.
    9. V. Ramaswami & Douglas Woolford & David Stanford, 2008. "The erlangization method for Markovian fluid flows," Annals of Operations Research, Springer, vol. 160(1), pages 215-225, April.
    10. Cheung, Eric C.K. & Zhu, Wei, 2023. "Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 84-101.
    11. Lesław Gajek & Marcin Rudź, 2018. "Risk-switching insolvency models," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 51, pages 129-146.
    12. Sarah Dendievel & Guy Latouche, 2017. "Approximations for Time-Dependent Distributions in Markovian Fluid Models," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 285-309, March.
    13. Alessandra Carleo & Mariafortuna Pietroluongo, 2014. "On matrix-exponential distributions in risk theory," Discussion Papers 2_2014, CRISEI, University of Naples "Parthenope", Italy.

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