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Two-sided Bounds for Renewal Equations and Ruin Quantities

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  • Stathis Chadjiconsatntinidis

    (University of Piraeus)

Abstract

In this paper, the objective is to provide sequences of improved non-increasing (non-decreasing) upper (lower) bounds for the solution of (defective) renewal equations in terms of the right-tail probability of a compound geometric distribution. Exponential (Lundberg type) and non-exponential type bounds are also derived. Also, under several reliability classifications, some new as well as improvements of well-known bounds are given. The results are then applied to obtain refinements of the bounds for several ruin related quantities, (such as the deficit at ruin, the joint distribution of the surplus prior to and at ruin, the mean deficit at ruin and the stop-loss premium, and the compound geometric densities). Bounds for the renewal function are also given.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:2:d:10.1007_s11009-024-10075-0
    DOI: 10.1007/s11009-024-10075-0
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    References listed on IDEAS

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    1. 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.
    2. Chadjiconstantinidis, Stathis & Xenos, Panos, 2022. "Refinements of bounds for tails of compound distributions and ruin probabilities," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    3. De Vylder, F. & Goovaerts, M., 1984. "Bounds for classical ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 3(2), pages 121-131, April.
    4. Jun Cai & José Garrido, 1999. "Two-Sided Bounds for Ruin Probabilities when the Adjustment Coefficient does not Exist," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 1999(1), pages 80-92.
    5. Willmot, Gordon E., 1997. "Bounds for compound distributions based on mean residual lifetimes and equilibrium distributions," Insurance: Mathematics and Economics, Elsevier, vol. 21(1), pages 25-42, October.
    6. 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.
    7. Lin, X. Sheldon & Willmot, Gordon E., 1999. "Analysis of a defective renewal equation arising in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 63-84, September.
    8. Willmot, Gordon E., 1994. "Refinements and distributional generalizations of Lundberg's inequality," Insurance: Mathematics and Economics, Elsevier, vol. 15(1), pages 49-63, October.
    9. Broeckx, F. & Goovaerts, M. & De Vylder, F., 1986. "Ordering of risks and ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 5(1), pages 35-39, January.
    10. Woo, Jae-Kyung, 2011. "Refinements of two-sided bounds for renewal equations," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 189-196, March.
    11. 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.
    12. 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.
    13. Psarrakos, Georgios, 2008. "Tail bounds for the distribution of the deficit in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 197-202, October.
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