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Ruin Probabilities and Deficit for the Renewal Risk Model with Phase-type Interarrival Times

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  • Avram, F.
  • Usábel, M.

Abstract

This paper shows how the multivariate finite time ruin probability function, in a phase-type environment, inherits the phase-type structure and can be efficiently approximated with only one Laplace transform inversion. From a theoretical point of view, we also provide below a generalization of Thorin’s formula (1971) for the double Laplace transform of the finite time ruin probability, by considering also the deficit at ruin; the model is that of a Sparre Andersen (renewal) risk process with phase-type interarrival times. In the case when the claims distribution is of phase-type as well, we obtain also an alternative formula for the single Laplace transform in time (or “exponentially killed probability’’), in terms of the roots with positive real part of the Lundberg’s equations, which complements Asmussen’s representation (1992) in terms of the roots with negative real part.

Suggested Citation

  • Avram, F. & Usábel, M., 2004. "Ruin Probabilities and Deficit for the Renewal Risk Model with Phase-type Interarrival Times," ASTIN Bulletin, Cambridge University Press, vol. 34(2), pages 315-332, November.
    • Handle: RePEc:cup:astinb:v:34:y:2004:i:02:p:315-332_01
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    Cited by:

    1. Li, Shuanming & Lu, Yi, 2009. "The distribution of total dividend payments in a Sparre Andersen model," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1246-1251, May.
    2. Ren, Jiandong, 2009. "A connection between the discounted and non-discounted expected penalty functions in the Sparre Andersen risk model," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 324-330, February.
    3. Yi Lu, 2016. "On the Evaluation of Expected Penalties at Claim Instants That Cause Ruin in the Classical Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 237-255, March.
    4. M. Concepcion Ausin & Michael P. Wiper & Rosa E. Lillo, 2009. "Bayesian estimation of finite time ruin probabilities," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(6), pages 787-805, November.
    5. Thampi K. K. & Jacob M. J. & Raju N., 2007. "Ruin Probabilities under Generalized Exponential Distribution," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 2(1), pages 1-12, May.
    6. Tang, Qihe & Wei, Li, 2010. "Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 19-31, February.
    7. Albrecher, Hansjorg & Boxma, Onno J., 2005. "On the discounted penalty function in a Markov-dependent risk model," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 650-672, December.

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