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The distributions of the time to reach a given level and the duration of negative surplus in the Erlang(2) risk model

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  • Dickson, David C.M.
  • Li, Shuanming

Abstract

We study the distributions of [1] the first time that the surplus reaches a given level and [2] the duration of negative surplus in a Sparre Andersen risk process with the inter-claim times being Erlang(2) distributed. These distributions can be obtained through the inversion of Laplace transforms using the inversion relationship for the Erlang(2) risk model given by Dickson and Li (2010).

Suggested Citation

  • Dickson, David C.M. & Li, Shuanming, 2013. "The distributions of the time to reach a given level and the duration of negative surplus in the Erlang(2) risk model," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 490-497.
    • Handle: RePEc:eee:insuma:v:52:y:2013:i:3:p:490-497
    DOI: 10.1016/j.insmatheco.2013.02.013
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    References listed on IDEAS

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    1. 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.
    2. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    3. 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.
    4. Shuanming Li, 2008. "The Time of Recovery and the Maximum Severity of Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 12(4), pages 413-425.
    5. Li, Shuanming & Garrido, Jose, 2004. "On ruin for the Erlang(n) risk process," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 391-408, June.
    6. Gerber, Hans U., 1990. "When does the surplus reach a given target?," Insurance: Mathematics and Economics, Elsevier, vol. 9(2-3), pages 115-119, September.
    7. Dickson,David C. M., 2010. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521176750.
    8. Sun, Li-Juan, 2005. "The expected discounted penalty at ruin in the Erlang (2) risk process," Statistics & Probability Letters, Elsevier, vol. 72(3), pages 205-217, May.
    9. Sundt, Bjørn & Jewell, William S., 1981. "Further Results on Recursive Evaluation of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 27-39, June.
    10. Dickson, David C.M. & Willmot, Gordon E., 2005. "The Density of the Time to Ruin in the Classical Poisson Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 45-60, May.
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    Cited by:

    1. Lanpeng Ji & Chunsheng Zhang, 2014. "A Duality Result for the Generalized Erlang Risk Model," Risks, MDPI, vol. 2(4), pages 1-11, November.
    2. Michael V. Boutsikas & Konstadinos Politis, 2017. "Exit Times, Overshoot and Undershoot for a Surplus Process in the Presence of an Upper Barrier," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 75-95, March.
    3. Landriault, David & Shi, Tianxiang, 2015. "Occupation times in the MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 75-82.
    4. Li, Yingqiu & Wei, Yushao & Peng, Zhaohui, 2021. "Occupation times for spectrally negative Lévy processes on the last exit time," Statistics & Probability Letters, Elsevier, vol. 175(C).
    5. 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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