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Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times

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  • Cheung, Eric C.K.

Abstract

In the context of a Sparre Andersen risk model with arbitrary interclaim time distribution, the moments of discounted aggregate claim costs until ruin are studied. Our analysis relies on a novel generalization of the so-called discounted density which further involves a moment-based component. More specifically, while the usual discounted density contains a discount factor with respect to the time of ruin, we propose to incorporate powers of the sum until ruin of the discounted (and possibly transformed) claims into the density. Probabilistic arguments are applied to derive defective renewal equations satisfied by the moments of discounted aggregate claim costs until ruin. Detailed examples concerning the discounted aggregate claims and the number of claims until ruin are studied upon assumption on the claim severities. Numerical illustrations are also given at the end.

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  • 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.
    • Handle: RePEc:eee:insuma:v:53:y:2013:i:2:p:343-354
    DOI: 10.1016/j.insmatheco.2013.06.003
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    References listed on IDEAS

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    15. Leveille, Ghislain & Garrido, Jose, 2001. "Moments of compound renewal sums with discounted claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(2), pages 217-231, April.
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    19. 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.
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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. Jae-Kyung Woo & Haibo Liu, 2018. "Discounted Aggregate Claim Costs Until Ruin in the Discrete-Time Renewal Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1285-1318, December.
    3. 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.
    4. Liu, Peng & Zhang, Chunsheng & Ji, Lanpeng, 2017. "A note on ruin problems in perturbed classical risk models," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 28-33.
    5. 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.
    6. Eric C.K. Cheung & Haibo Liu & Jae-Kyung Woo, 2015. "On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy," Risks, MDPI, vol. 3(4), pages 1-24, November.
    7. Jiechang Ruan & Wenguang Yu & Ke Song & Yihan Sun & Yujuan Huang & Xinliang Yu, 2019. "A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model," Mathematics, MDPI, vol. 7(10), pages 1-12, September.
    8. 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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