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On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model

Author

Listed:
  • Wenguang Yu

    (School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China)

  • Peng Guo

    (College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China)

  • Qi Wang

    (School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China)

  • Guofeng Guan

    (School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China)

  • Qing Yang

    (School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China)

  • Yujuan Huang

    (School of Science, Shandong Jiaotong University, Jinan 250357, China)

  • Xinliang Yu

    (School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China)

  • Boyi Jin

    (School of Insurance, Shandong University of Finance and Economics, Jinan 250014, China)

  • Chaoran Cui

    (School of Computer Science & Technology, Shandong University of Finance and Economics, Jinan 250014, China)

Abstract

In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical risk model. We derive the equations and the boundary conditions satisfied by the Gerber-Shiu function, the expected discounted capital injection function and the expected discounted dividend function by assuming that the observation interval and claim amount are exponentially distributed, respectively. Numerical examples are also given to further analyze the influence of relevant parameters on the actuarial function of the risk model.

Suggested Citation

  • Wenguang Yu & Peng Guo & Qi Wang & Guofeng Guan & Qing Yang & Yujuan Huang & Xinliang Yu & Boyi Jin & Chaoran Cui, 2020. "On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model," Mathematics, MDPI, vol. 8(4), pages 1-21, April.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:511-:d:340681
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    References listed on IDEAS

    as
    1. Zhao, Yongxia & Chen, Ping & Yang, Hailiang, 2017. "Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 135-146.
    2. Yin, Chuancun & Wen, Yuzhen & Zhao, Yongxia, 2014. "On The Optimal Dividend Problem For A Spectrally Positive Lã‰Vy Process," ASTIN Bulletin, Cambridge University Press, vol. 44(3), pages 635-651, September.
    3. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    4. Yuen, Kam C. & Wang, Guojing & Li, Wai K., 2007. "The Gerber-Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier," Insurance: Mathematics and Economics, Elsevier, vol. 40(1), pages 104-112, January.
    5. Yang, Yang & Su, Wen & Zhang, Zhimin, 2019. "Estimating the discounted density of the deficit at ruin by Fourier cosine series expansion," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 147-155.
    6. Zeng, Yan & Li, Danping & Gu, Ailing, 2016. "Robust equilibrium reinsurance-investment strategy for a mean–variance insurer in a model with jumps," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 138-152.
    7. Xu, Ran & Woo, Jae-Kyung & Han, Xixuan & Yang, Hailiang, 2018. "A plan of capital injections based on the claims frequency," Annals of Actuarial Science, Cambridge University Press, vol. 12(2), pages 296-325, September.
    8. Zhang, Zhimin & Han, Xiao, 2017. "The compound Poisson risk model under a mixed dividend strategy," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 1-12.
    9. Zhang, Zhimin & Cheung, Eric C.K. & Yang, Hailiang, 2018. "On The Compound Poisson Risk Model With Periodic Capital Injections," ASTIN Bulletin, Cambridge University Press, vol. 48(1), pages 435-477, January.
    10. Zeng, Yan & Li, Danping & Chen, Zheng & Yang, Zhou, 2018. "Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility," Journal of Economic Dynamics and Control, Elsevier, vol. 88(C), pages 70-103.
    11. Wenguang Yu, 2013. "Randomized Dividends in a Discrete Insurance Risk Model with Stochastic Premium Income," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-9, March.
    12. Yujuan Huang & Wenguang Yu, 2014. "The Gerber-Shiu Discounted Penalty Function of Sparre Andersen Risk Model with a Constant Dividend Barrier," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-7, July.
    13. Zhimin Zhang & Eric C. K. Cheung, 2016. "The Markov Additive Risk Process Under an Erlangized Dividend Barrier Strategy," Methodology and Computing in Applied Probability, Springer, vol. 18(2), pages 275-306, June.
    14. Albrecher, Hansjörg & Cheung, Eric C.K. & Thonhauser, Stefan, 2011. "Randomized Observation Periods for the Compound Poisson Risk Model: Dividends," ASTIN Bulletin, Cambridge University Press, vol. 41(2), pages 645-672, November.
    15. Chi, Yichun & Lin, X. Sheldon, 2011. "On the threshold dividend strategy for a generalized jump-diffusion risk model," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 326-337, May.
    16. Sheldon Lin, X. & E. Willmot, Gordon & Drekic, Steve, 2003. "The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 551-566, December.
    17. 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.
    18. Choi, Michael C.H. & Cheung, Eric C.K., 2014. "On the expected discounted dividends in the Cramér–Lundberg risk model with more frequent ruin monitoring than dividend decisions," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 121-132.
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