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Integrated insurance risk models with exponential Lévy investment

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  • Klüppelberg, Claudia
  • Kostadinova, Radostina
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    Abstract

    We consider an insurance risk model for the cashflow of an insurance company, which invests its reserve into a portfolio consisting of risky and riskless assets. The price of the risky asset is modeled by an exponential Lévy process. We derive the integrated risk process and the corresponding discounted net loss process. We calculate certain quantities as characteristic functions and moments. We also show under weak conditions stationarity of the discounted net loss process and derive the left and right tail behavior of the model. Our results show that the model carries a high risk, which may originate either from large insurance claims or from the risky investment.

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    Bibliographic Info

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

    Volume (Year): 42 (2008)
    Issue (Month): 2 (April)
    Pages: 560-577

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    Handle: RePEc:eee:insuma:v:42:y:2008:i:2:p:560-577

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

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    1. Paulsen, Jostein, 1993. "Risk theory in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 46(2), pages 327-361, June.
    2. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, University of Chicago Press, vol. 63(4), pages 511-24, October.
    3. Lindner, Alexander & Maller, Ross, 2005. "Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 115(10), pages 1701-1722, October.
    4. Anna Frolova & Serguei Pergamenshchikov & Yuri Kabanov, 2002. "In the insurance business risky investments are dangerous," Finance and Stochastics, Springer, Springer, vol. 6(2), pages 227-235.
    5. Susanne Emmer & Claudia Klüppelberg & Ralf Korn, 2001. "Optimal Portfolios with Bounded Capital at Risk," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 11(4), pages 365-384.
    6. Hipp, Christian & Plum, Michael, 2000. "Optimal investment for insurers," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 27(2), pages 215-228, October.
    7. Paulsen, Jostein, 1998. "Ruin theory with compounding assets -- a survey," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 22(1), pages 3-16, May.
    8. Kostadinova, Radostina, 2007. "Optimal investment for insurers when the stock price follows an exponential Lévy process," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 41(2), pages 250-263, September.
    9. Tang, Qihe & Tsitsiashvili, Gurami, 2003. "Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 108(2), pages 299-325, December.
    10. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 92(2), pages 265-285, April.
    11. Nilsen, Trygve & Paulsen, Jostein, 1996. "On the distribution of a randomly discounted compound Poisson process," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 61(2), pages 305-310, February.
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    Cited by:
    1. Tang, Qihe & Wang, Guojing & Yuen, Kam C., 2010. "Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, Elsevier, vol. 46(2), pages 362-370, April.

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