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Tail bounds for the joint distribution of the surplus prior to and at ruin

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  • Psarrakos, Georgios
  • Politis, Konstadinos

Abstract

For the classical risk model with Poisson arrivals, we study the (bivariate) tail of the joint distribution of the surplus prior to and at ruin. We obtain some exact expressions and new bounds for this tail, and we suggest three numerical methods that may yield upper and lower bounds for it. As a by-product of the analysis, we obtain new upper and lower bounds for the probability and severity of ruin. Many of the bounds in the present paper improve and generalise corresponding bounds that have appeared earlier. For the numerical bounds, their performance is also compared against bounds available in the literature.

Suggested Citation

  • Psarrakos, Georgios & Politis, Konstadinos, 2008. "Tail bounds for the joint distribution of the surplus prior to and at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 163-176, February.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:163-176
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    References listed on IDEAS

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    1. Dickson, David C. M., 1992. "On the distribution of the surplus prior to ruin," Insurance: Mathematics and Economics, Elsevier, vol. 11(3), pages 191-207, October.
    2. Dickson, David C.M. & dos Reis, Alfredo D. Egídio & Waters, Howard R., 1995. "Some Stable Algorithms in Ruin Theory and Their Applications," ASTIN Bulletin, Cambridge University Press, vol. 25(2), pages 153-175, November.
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    8. Dickson, D. C. M., 2001. "Lundberg Approximations for Compound Distributions with Insurance Applications. By G. E. Willmot and X. S. Lin. (Springer, 2000)," British Actuarial Journal, Cambridge University Press, vol. 7(4), pages 690-691, October.
    9. Politis, Konstadinos, 2005. "Bounds for the probability and severity of ruin in the Sparre Andersen model," Insurance: Mathematics and Economics, Elsevier, vol. 36(2), pages 165-177, April.
    10. Chadjiconstantinidis, Stathis & Politis, Konstadinos, 2007. "Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 41-52, July.
    11. Ng, Andrew C.Y. & Yang, Hailiang, 2005. "Lundberg-type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin under a Markov-modulated Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 35(2), pages 351-361, November.
    12. Andrew Ng & Hailiang Yang, 2005. "Lundberg-Type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin Under the Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 85-100.
    13. Willmot, Gordon E., 2002. "Compound geometric residual lifetime distributions and the deficit at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 30(3), pages 421-438, June.
    14. Cai, Jun & Garrido, Jose, 1998. "Aging properties and bounds for ruin probabilities and stop-loss premiums," Insurance: Mathematics and Economics, Elsevier, vol. 23(1), pages 33-43, October.
    15. Gerber, Hans U. & Shiu, Elias S. W., 1997. "The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 129-137, November.
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    Cited by:

    1. Lazaros Kanellopoulos, 2023. "Some Stochastic Orders over an Interval with Applications," Risks, MDPI, vol. 11(9), pages 1-14, September.
    2. Woo, Jae-Kyung, 2011. "Refinements of two-sided bounds for renewal equations," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 189-196, March.
    3. Stathis Chadjiconsatntinidis, 2024. "Two-sided Bounds for Renewal Equations and Ruin Quantities," Methodology and Computing in Applied Probability, Springer, vol. 26(2), pages 1-54, June.
    4. 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.
    5. Psarrakos, Georgios, 2009. "Asymptotic results for heavy-tailed distributions using defective renewal equations," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 774-779, March.

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