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On the analysis of ruin-related quantities in the delayed renewal risk model

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  • Kim, So-Yeun
  • Willmot, Gordon E.

Abstract

This paper first explores the Laplace transform of the time of ruin in the delayed renewal risk model. We show that Ḡδd(u), the Laplace transform of the time of ruin in the delayed model, also satisfies a defective renewal equation and use this to study the Cramer–Lundberg asymptotics and bounds of Ḡδd(u). Next, the paper considers the stochastic decomposition of the residual lifetime of maximal aggregate loss and more generally Lδd in the delayed renewal risk model, using the framework equation introduced in Kim and Willmot (2011) and the defective renewal equation for the Laplace transform of the time of ruin. As a result of the decomposition, we propose a way to calculate the mean and the moments of the proper deficit in the delayed renewal risk model. Lastly, closed form expressions are derived for the Gerber–Shiu function in the delayed renewal risk model with the distributional assumption of time until the first claim and simulation results are included to assess the impact of different distributional assumptions on the time until the first claim.

Suggested Citation

  • Kim, So-Yeun & Willmot, Gordon E., 2016. "On the analysis of ruin-related quantities in the delayed renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 77-85.
    • Handle: RePEc:eee:insuma:v:66:y:2016:i:c:p:77-85
    DOI: 10.1016/j.insmatheco.2015.10.011
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    References listed on IDEAS

    as
    1. Woo, Jae-Kyung, 2010. "Some Remarks on Delayed Renewal Risk Models," ASTIN Bulletin, Cambridge University Press, vol. 40(1), pages 199-219, May.
    2. Willmot, Gordon E., 2007. "On the discounted penalty function in the renewal risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 17-31, July.
    3. David Landriault & Gordon Willmot, 2009. "On the Joint Distributions of the Time to Ruin, the Surplus Prior to Ruin, and the Deficit at Ruin in the Classical Risk Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(2), pages 252-270.
    4. Willmot, Gordon E., 2004. "A note on a class of delayed renewal risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 34(2), pages 251-257, April.
    5. Willmot, Gordon E. & Dickson, David C. M., 2003. "The Gerber-Shiu discounted penalty function in the stationary renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 403-411, July.
    6. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    7. Dickson, D. C. M., 2001. "Lundberg Approximations for Compound Distributions with Insurance Applications. By G. E. Willmot and X. S. Lin. (Springer, 2000)," British Actuarial Journal, Cambridge University Press, vol. 7(4), pages 690-691, October.
    8. Dickson, David C. M. & Hipp, Christian, 2001. "On the time to ruin for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 333-344, December.
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