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Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence

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  • Tang, Qihe
  • Wei, Li

Abstract

We study the asymptotic behavior of the Gerber-Shiu expected discounted penalty function in the renewal risk model. Under the assumption that the claim-size distribution has a convolution-equivalent density function, which allows both heavy-tailed and light-tailed cases, we establish some asymptotic formulas for the Gerber-Shiu function with a fairly general penalty function. These formulas become completely transparent in the compound Poisson risk model or for certain choices of the penalty function in the renewal risk model. A by-product of this work is an extension of the Wiener-Hopf factorization to include the times of ascending and descending ladders in the continuous-time renewal risk model.

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  • 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.
    • Handle: RePEc:eee:insuma:v:46:y:2010:i:1:p:19-31
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Albrecher, Hansjoerg & Constantinescu, Corina & Thomann, Enrique, 2012. "Asymptotic results for renewal risk models with risky investments," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3767-3789.
    5. Griffin, Philip S. & Maller, Ross A. & Roberts, Dale, 2013. "Finite time ruin probabilities for tempered stable insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 478-489.
    6. Wang, Kaiyong & Yang, Yang & Yu, Changjun, 2013. "Estimates for the overshoot of a random walk with negative drift and non-convolution equivalent increments," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1504-1512.
    7. Geiger Daniel J. & Adekpedjou Akim, 2019. "On corrected phase-type approximations of the time value of ruin with heavy tails," Statistics & Risk Modeling, De Gruyter, vol. 36(1-4), pages 57-75, December.
    8. Lefèvre, Claude & Trufin, Julien & Zuyderhoff, Pierre, 2017. "Some comparison results for finite-time ruin probabilities in the classical risk model," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 143-149.
    9. Griffin, Philip S. & Maller, Ross A. & Schaik, Kees van, 2012. "Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 382-392.

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