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Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models

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  • Cheung, Eric C.K.
  • Landriault, David
  • Willmot, Gordon E.
  • Woo, Jae-Kyung

Abstract

The structure of various Gerber-Shiu functions in Sparre Andersen models allowing for possible dependence between claim sizes and interclaim times is examined. The penalty function is assumed to depend on some or all of the surplus immediately prior to ruin, the deficit at ruin, the minimum surplus before ruin, and the surplus immediately after the second last claim before ruin. Defective joint and marginal distributions involving these quantities are derived. Many of the properties in the Sparre Andersen model without dependence are seen to hold in the present model as well. A discussion of Lundberg's fundamental equation and the generalized adjustment coefficient is given, and the connection to a defective renewal equation is considered. The usual Sparre Andersen model without dependence is also discussed, and in particular the case with exponential claim sizes is considered.

Suggested Citation

  • Cheung, Eric C.K. & Landriault, David & Willmot, Gordon E. & Woo, Jae-Kyung, 2010. "Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 117-126, February.
    • Handle: RePEc:eee:insuma:v:46:y:2010:i:1:p:117-126
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    References listed on IDEAS

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    Cited by:

    1. Heilpern, Stanislaw, 2014. "Ruin measures for a compound Poisson risk model with dependence based on the Spearman copula and the exponential claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 251-257.
    2. Zan Yu & Lianzeng Zhang, 2024. "Computing the Gerber-Shiu function with interest and a constant dividend barrier by physics-informed neural networks," Papers 2401.04378, arXiv.org.
    3. Lee, Wing Yan & Willmot, Gordon E., 2014. "On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 1-10.
    4. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    5. Dila Puspita & Adam Kolkiewicz & Ken Seng Tan, 2020. "Discrete Time Ruin Probability for Takaful (Islamic Insurance) with Investment and Qard-Hasan (Benevolent Loan) Activities," JRFM, MDPI, vol. 13(9), pages 1-24, September.
    6. Cheung, Eric C.K. & Feng, Runhuan, 2013. "A unified analysis of claim costs up to ruin in a Markovian arrival risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 98-109.
    7. Boudreault, Mathieu & Cossette, Hélène & Marceau, Étienne, 2014. "Risk models with dependence between claim occurrences and severities for Atlantic hurricanes," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 123-132.
    8. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    9. Ivanovs, Jevgenijs, 2013. "A note on killing with applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 52(1), pages 29-34.
    10. Landriault, David & Shi, Tianxiang & Willmot, Gordon E., 2011. "Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 371-379.
    11. Woo, Jae-Kyung & Cheung, Eric C.K., 2013. "A note on discounted compound renewal sums under dependency," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 170-179.
    12. Orbán Mihálykó, Éva & Mihálykó, Csaba, 2011. "Mathematical investigation of the Gerber-Shiu function in the case of dependent inter-claim time and claim size," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 378-383, May.
    13. Cheung, Eric C.K., 2013. "Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 343-354.
    14. Landriault, David & Li, Bin & Loke, Sooie-Hoe & Willmot, Gordon E. & Xu, Di, 2017. "A note on the convexity of ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 1-6.
    15. Cheung, Eric C.K. & Liu, Haibo & Willmot, Gordon E., 2018. "Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 358-377.
    16. Cheung, Eric C.K., 2011. "A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 384-397, May.
    17. Sung Soo Kim & Steve Drekic, 2016. "Ruin Analysis of a Discrete-Time Dependent Sparre Andersen Model with External Financial Activities and Randomized Dividends," Risks, MDPI, vol. 4(1), pages 1-15, February.
    18. 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.
    19. Franck Adékambi & Essodina Takouda, 2022. "On the Discounted Penalty Function in a Perturbed Erlang Renewal Risk Model With Dependence," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 481-513, June.
    20. Willmot, Gordon E. & Woo, Jae-Kyung, 2012. "On the analysis of a general class of dependent risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 134-141.
    21. Zhimin Zhang & Hailiang Yang & Hu Yang, 2012. "On a Sparre Andersen Risk Model with Time-Dependent Claim Sizes and Jump-Diffusion Perturbation," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 973-995, December.
    22. Cossette, Hélène & Marceau, Etienne & Trufin, Julien & Zuyderhoff, Pierre, 2020. "Ruin-based risk measures in discrete-time risk models," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 246-261.
    23. Wong, Jeff T.Y. & Cheung, Eric C.K., 2015. "On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 280-290.

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