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Numerical Method for a Perturbed Risk Model with Proportional Investment

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  • Chunwei Wang

    (School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China
    These authors contributed equally to this work.)

  • Naidan Deng

    (School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China
    These authors contributed equally to this work.)

  • Silian Shen

    (School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China)

Abstract

In this paper, we study the perturbed risk model with a threshold dividend strategy and proportional investment. The insurance companies are allowed to invest their surplus in a financial market consisting of a risk-free asset and a risky asset in fixed proportions; the risky assets are modeled by the jump-diffusion process. Firstly, using the theory of the stochastic process and stochastic analysis, we obtained the integro-differential equations satisfied by the expected discounted dividend payments and the discounted penalty function. Secondly, we obtained the numerical approximate solutions of the integro-differential equations through the sinc method, since the analytical solutions of them are not easy to obtain, and we found that the error is within a manageable range. Finally, we considered some numerical examples where the claim sizes follow an exponential distribution, a mixture of two exponential distributions or the lognormal distribution in detail, and explored how perturbations and proportional investment affect dividends and ruin probability. Moreover, sensitive analysis showed that the proportion of the risky investment, the diffusion coefficient, the distribution of the claims and the positive jump in the risky assets investment all have explicit impacts on dividends and ruin probability.

Suggested Citation

  • Chunwei Wang & Naidan Deng & Silian Shen, 2022. "Numerical Method for a Perturbed Risk Model with Proportional Investment," Mathematics, MDPI, vol. 11(1), pages 1-27, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2022:i:1:p:43-:d:1011769
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    References listed on IDEAS

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    1. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    2. Chen, Xu & Xiao, Ting & Yang, Xiang-qun, 2014. "A Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 76-83.
    3. Zhang, Zhimin, 2017. "Approximating The Density Of The Time To Ruin Via Fourier-Cosine Series Expansion," ASTIN Bulletin, Cambridge University Press, vol. 47(1), pages 169-198, January.
    4. Priscilla Serwaa Nkyira Gambrah & Traian Adrian Pirvu, 2014. "Risk Measures and Portfolio Optimization," JRFM, MDPI, vol. 7(3), pages 1-17, September.
    5. Chi, Yichun, 2010. "Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 385-396, April.
    6. Yujuan Huang & Jing Li & Hengyu Liu & Wenguang Yu, 2021. "Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method," Mathematics, MDPI, vol. 9(9), pages 1-17, April.
    7. Wan, Ning, 2007. "Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 509-523, May.
    8. Vierkötter, Matthias & Schmidli, Hanspeter, 2017. "On optimal dividends with exponential and linear penalty payments," Insurance: Mathematics and Economics, Elsevier, vol. 72(C), pages 265-270.
    9. Jiaqi Zhu & Shenghong Li, 2020. "Time-Consistent Investment and Reinsurance Strategies for Mean-Variance Insurers under Stochastic Interest Rate and Stochastic Volatility," Mathematics, MDPI, vol. 8(12), pages 1-22, December.
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