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Analysis of the Gerber-Shiu function and dividend barrier problems for a risk process with two classes of claims

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  • Chadjiconstantinidis, Stathis
  • Papaioannou, Apostolos D.
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    Abstract

    In this paper we consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, the Poisson and the generalized Erlang(2) process. We prove that the Gerber-Shiu function satisfies some defective renewal equations. Exact representations for the solutions of these equations are derived through an associated compound geometric distribution and an analytic expression for this quantity is given when the claim severities have rationally distributed Laplace transforms. Further, the same risk model is considered in the presence of a constant dividend barrier. A system of integro-differential equations with certain boundary conditions for the Gerber-Shiu function is derived and solved. Using systems of integro-differential equations for the moment-generating function as well as for the arbitrary moments of the discounted sum of the dividend payments until ruin, a matrix version of the dividends-penalty is derived. An extension to a risk model when the two independent claim counting processes are Poisson and generalized Erlang([nu]), respectively, is considered, generalizing the aforementioned results.

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    Bibliographic Info

    Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

    Volume (Year): 45 (2009)
    Issue (Month): 3 (December)
    Pages: 470-484

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    Handle: RePEc:eee:insuma:v:45:y:2009:i:3:p:470-484

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    Web page: http://www.elsevier.com/locate/inca/505554

    Related research

    Keywords: Compound Poisson process Generalized Erlang risk process Discounted penalty function Defective renewal equations Dividend barrier Rationally distributed claim severities Present value of the dividend payments Moment-generating function Dividends-penalty identity;

    References

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    1. Tsai, Cary Chi-Liang & Sun, Li-juan, 2004. "On the discounted distribution functions for the Erlang(2) risk process," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 35(1), pages 5-19, August.
    2. Li, Shuanming & Garrido, Jose, 2004. "On ruin for the Erlang(n) risk process," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 34(3), pages 391-408, June.
    3. Lin, X. Sheldon & Willmot, Gordon E., 1999. "Analysis of a defective renewal equation arising in ruin theory," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 25(1), pages 63-84, September.
    4. Lu, Yi & Li, Shuanming, 2009. "The Markovian regime-switching risk model with a threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 44(2), pages 296-303, April.
    5. Li, Shuanming & Dickson, David C.M., 2006. "The maximum surplus before ruin in an Erlang(n) risk process and related problems," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 38(3), pages 529-539, June.
    6. Li, Shuanming & Garrido, Jose, 2004. "On a class of renewal risk models with a constant dividend barrier," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 35(3), pages 691-701, December.
    7. Li, Shuanming & Lu, Yi, 2005. "On the expected discounted penalty functions for two classes of risk processes," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 36(2), pages 179-193, April.
    8. Dickson, David C. M. & Hipp, Christian, 2001. "On the time to ruin for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 29(3), pages 333-344, December.
    9. Dickson, David C. M. & Hipp, Christian, 1998. "Ruin probabilities for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 22(3), pages 251-262, July.
    10. Yuen, Kam C. & Guo, Junyi & Wu, Xueyuan, 2002. "On a correlated aggregate claims model with Poisson and Erlang risk processes," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 31(2), pages 205-214, October.
    11. Albrecher, Hansjorg & Claramunt, M.Merce & Marmol, Maite, 2005. "On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 37(2), pages 324-334, October.
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