The expected discounted penalty at ruin in the Erlang (2) risk process
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.
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Volume (Year): 72 (2005)
Issue (Month): 3 (May)
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- 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.
- 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.
- 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.
- 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.
- 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.
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