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The expected discounted penalty at ruin in the Erlang (2) risk process


  • Sun, Li-Juan


In this paper, under the Erlang (2) risk process, we examine the expected discounted value of a penalty at ruin, which is considered as a function of the initial surplus. We first show that the expected discounted penalty function satisfies an integro-differential equation, and give its initial value, as well as its Laplace transform. We further prove that this function is twice differentiable, and satisfies a defective renewal equation. An explicit expression for the solution of this equation can be derived. The associated compound geometric distribution and "claim size" distribution are also studied.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:stapro:v:72:y:2005:i:3:p:205-217

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    References listed on IDEAS

    1. Cai, Jun & Dickson, David C. M., 2002. "On the expected discounted penalty function at ruin of a surplus process with interest," Insurance: Mathematics and Economics, Elsevier, vol. 30(3), pages 389-404, June.
    2. 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.
    3. 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.
    4. Lin, X. Sheldon & Willmot, Gordon E., 1999. "Analysis of a defective renewal equation arising in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 63-84, September.
    5. 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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    Cited by:

    1. 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.
    2. Huang, Tao & Zhao, Ruiqing & Tang, Wansheng, 2009. "Risk model with fuzzy random individual claim amount," European Journal of Operational Research, Elsevier, vol. 192(3), pages 879-890, February.
    3. Thampi K. K. & Jacob M. J. & Raju N., 2007. "Ruin Probabilities under Generalized Exponential Distribution," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 2(1), pages 1-12, May.


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