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The time to ruin and the number of claims until ruin for phase-type claims

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  • Frostig, Esther
  • Pitts, Susan M.
  • Politis, Konstadinos

Abstract

We consider a renewal risk model with phase-type claims, and obtain an explicit expression for the joint transform of the time to ruin and the number of claims until ruin, with a penalty function applied to the deficit at ruin. The approach is via the duality between a risk model with phase-type claims and a particular single server queueing model with phase-type customer interarrival times; see Frostig (2004). This result specializes to one for the probability generating function of the number of claims until ruin. We obtain explicit expressions for the distribution of the number of claims until ruin for exponentially distributed claims when the inter-claim times have an Erlang-n distribution.

Suggested Citation

  • Frostig, Esther & Pitts, Susan M. & Politis, Konstadinos, 2012. "The time to ruin and the number of claims until ruin for phase-type claims," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 19-25.
  • Handle: RePEc:eee:insuma:v:51:y:2012:i:1:p:19-25
    DOI: 10.1016/j.insmatheco.2012.02.013
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    References listed on IDEAS

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    1. Borovkov, Konstantin A. & Dickson, David C.M., 2008. "On the ruin time distribution for a Sparre Andersen process with exponential claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1104-1108, June.
    2. Dickson, David C.M. & Li, Shuanming, 2010. "Finite time ruin problems for the Erlang(2) risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 12-18, February.
    3. 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.
    4. Dickson,David C. M., 2010. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521176750.
    5. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    6. Hesselager, Ole, 1994. "A Recursive Procedure for Calculation of some Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 24(1), pages 19-32, May.
    7. Landriault, David & Willmot, Gordon, 2008. "On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 600-608, April.
    8. Egidio dos Reis, Alfredo D., 2002. "How many claims does it take to get ruined and recovered?," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 235-248, October.
    9. Ahn, Soohan & Badescu, Andrei L., 2007. "On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 234-249, September.
    10. Panjer, H. H. & Willmot, G. E., 1982. "Recursions for Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 13(1), pages 1-12, June.
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    Cited by:

    1. 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.
    2. Boxma, Onno & Frostig, Esther & Perry, David & Yosef, Rami, 2017. "A state dependent reinsurance model," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 170-181.
    3. 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.
    4. Esther Frostig & Adva Keren-Pinhasik, 2020. "Parisian Ruin with Erlang Delay and a Lower Bankruptcy Barrier," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 101-134, March.
    5. Avram, F. & Pistorius, M., 2014. "On matrix exponential approximations of ruin probabilities for the classic and Brownian perturbed Cramér–Lundberg processes," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 57-64.
    6. Philipp Lukas Strietzel & Anita Behme, 2022. "Moments of the Ruin Time in a Lévy Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 3075-3099, December.
    7. 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.
    8. 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.
    9. 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.

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