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The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest

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  • Wang, Guojing
  • Wu, Rong

Abstract

In this paper, we consider the Gerber-Shiu expected discounted penalty function for the perturbed compound Poisson risk process with constant force of interest. We decompose the Gerber-Shiu function into two parts: the expected discounted penalty at ruin that is caused by a claim and the expected discounted penalty at ruin due to oscillation. We derive the integral equations and the integro-differential equations for them. By solving the integro-differential equations we get some closed form expressions for the expected discounted penalty functions under certain assumptions.

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  • Wang, Guojing & Wu, Rong, 2008. "The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 59-64, February.
    • Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:59-64
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    References listed on IDEAS

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    1. Wang, Guojing & Wu, Rong, 2001. "Distributions for the risk process with a stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 95(2), pages 329-341, October.
    2. Wang, Guojing, 2001. "A decomposition of the ruin probability for the risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 28(1), pages 49-59, February.
    3. Cai, Jun, 2004. "Ruin probabilities and penalty functions with stochastic rates of interest," Stochastic Processes and their Applications, Elsevier, vol. 112(1), pages 53-78, July.
    4. Hans Gerber & Elias Shiu, 2005. "The Time Value of Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 49-69.
    5. Schmidli, H., 1995. "Cramer-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 16(2), pages 135-149, May.
    6. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    7. Tsai, Cary Chi-Liang, 2003. "On the expectations of the present values of the time of ruin perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 413-429, July.
    8. Tsai, Cary Chi-Liang & Willmot, Gordon E., 2002. "On the moments of the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 327-350, December.
    9. Chiu, S. N. & Yin, C. C., 2003. "The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 59-66, August.
    10. Furrer, H. J. & Schmidli, H., 1994. "Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 15(1), pages 23-36, October.
    11. Wang, Guojing & Wu, Rong, 2000. "Some distributions for classical risk process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 15-24, February.
    12. Gerber, Hans U. & Landry, Bruno, 1998. "On the discounted penalty at ruin in a jump-diffusion and the perpetual put option," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 263-276, July.
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    Cited by:

    1. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    2. Yin, Chuancun & Wen, Yuzhen, 2013. "An extension of Paulsen–Gjessing’s risk model with stochastic return on investments," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 469-476.
    3. Wei Wang, 2015. "The Perturbed Sparre Andersen Model with Interest and a Threshold Dividend Strategy," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 251-283, June.
    4. Yu, Wenguang, 2013. "Some results on absolute ruin in the perturbed insurance risk model with investment and debit interests," Economic Modelling, Elsevier, vol. 31(C), pages 625-634.
    5. Yue He & Reiichiro Kawai & Yasutaka Shimizu & Kazutoshi Yamazaki, 2022. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Papers 2203.10680, arXiv.org, revised Dec 2022.
    6. Diko, Peter & Usábel, Miguel, 2011. "A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion process," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 126-131, July.
    7. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    8. Chuancun Yin & Yuzhen Wen, 2013. "An extension of Paulsen-Gjessing's risk model with stochastic return on investments," Papers 1302.6757, arXiv.org.

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