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Extended Gerber–Shiu functions in a risk model with interest

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  • Schmidli, Hanspeter

Abstract

We consider a compound Poisson risk model with interest. The Gerber–Shiu discounted penalty function is modified with an additional penalty for reaching a level above the initial capital. We show that the problem can be split into two independent problems; an original Gerber–Shiu function and a first passage problem. We also consider the case of negative interest. Finally, we apply the results to a model considered by Embrechts and Schmidli (1994).

Suggested Citation

  • Schmidli, Hanspeter, 2015. "Extended Gerber–Shiu functions in a risk model with interest," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 271-275.
    • Handle: RePEc:eee:insuma:v:61:y:2015:i:c:p:271-275
    DOI: 10.1016/j.insmatheco.2015.01.012
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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. Sundt, Bjorn & Teugels, Jozef L., 1995. "Ruin estimates under interest force," Insurance: Mathematics and Economics, Elsevier, vol. 16(1), pages 7-22, April.
    3. Schmidli, Hanspeter, 1999. "On the Distribution of the Surplus Prior and at Ruin," ASTIN Bulletin, Cambridge University Press, vol. 29(2), pages 227-244, November.
    4. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    5. Gerber, Hans U. & Goovaerts, Marc J. & Kaas, Rob, 1987. "On the Probability and Severity of Ruin," ASTIN Bulletin, Cambridge University Press, vol. 17(2), pages 151-163, November.
    6. Dufresne, Francois & Gerber, Hans U., 1988. "The surpluses immediately before and at ruin, and the amount of the claim causing ruin," Insurance: Mathematics and Economics, Elsevier, vol. 7(3), pages 193-199, October.
    7. Gerber, Hans U. & Lin, X. Sheldon & Yang, Hailiang, 2006. "A Note on the Dividends-Penalty Identity and the Optimal Dividend Barrier," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 489-503, November.
    8. Jun Cai & Runhuan Feng & Gordon E. Willmot, 2009. "The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber–Shiu Discounted Penalty Function," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 401-423, September.
    9. Hailiang Yang & Lihong Zhang, 2001. "The Joint Distribution of Surplus Immediately before Ruin and the Deficit at Ruin under Interest Force," North American Actuarial Journal, Taylor & Francis Journals, vol. 5(3), pages 92-103.
    10. Yang, Hailiang & Zhang, Lihong, 2001. "On the distribution of surplus immediately before ruin under interest force," Statistics & Probability Letters, Elsevier, vol. 55(3), pages 329-338, December.
    11. Cai, Jun & Feng, Runhuan & Willmot, Gordon E., 2009. "Analysis of the Compound Poisson Surplus Model with Liquid Reserves, Interest and Dividends," ASTIN Bulletin, Cambridge University Press, vol. 39(1), pages 225-247, May.
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    Cited by:

    1. Sooie-Hoe Loke & Enrique Thomann, 2018. "Numerical Ruin Probability in the Dual Risk Model with Risk-Free Investments," Risks, MDPI, vol. 6(4), pages 1-13, October.
    2. 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.
    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. 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.

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