IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i12p1402-d576424.html
   My bibliography  Save this article

Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

Author

Listed:
  • Wen Su

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

  • Yunyun Wang

    (Department of Econometrics and Business Statistics, School of Business, Monash University, Melbourne 3800, Australia)

Abstract

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite.

Suggested Citation

  • Wen Su & Yunyun Wang, 2021. "Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion," Mathematics, MDPI, vol. 9(12), pages 1-18, June.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:12:p:1402-:d:576424
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/12/1402/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/9/12/1402/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Yuan Gao & Lingju Chen & Jiancheng Jiang & Honglong You, 2020. "Nonparametric Estimation of the Ruin Probability in the Classical Compound Poisson Risk Model," JRFM, MDPI, vol. 13(12), pages 1-12, November.
    2. Zhang, Zhimin & Yang, Hailiang, 2013. "Nonparametric estimate of the ruin probability in a pure-jump Lévy risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 24-35.
    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. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    5. Sun, Xianming & Gan, Siqing & Vanmaele, Michèle, 2015. "Analytical approximation for distorted expectations," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 246-252.
    6. Marjon Ruijter & Kees Oosterlee, 2012. "Two-dimensional Fourier cosine series expansion method for pricing financial options," CPB Discussion Paper 225.rdf, CPB Netherlands Bureau for Economic Policy Analysis.
    7. Zhang, Zhimin & Yang, Hailiang, 2014. "Nonparametric estimation for the ruin probability in a Lévy risk model under low-frequency observation," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 168-177.
    8. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    9. Luca Di Persio & Gregorio Pellegrini & Michele Bonollo, 2015. "Polynomial Chaos Expansion Approach to Interest Rate Models," Journal of Probability and Statistics, Hindawi, vol. 2015, pages 1-24, December.
    10. Xie, Jiayi & Zhang, Zhimin, 2021. "Finite-time dividend problems in a Lévy risk model under periodic observation," Applied Mathematics and Computation, Elsevier, vol. 398(C).
    11. Marjon Ruijter & Kees Oosterlee, 2012. "Two-dimensional Fourier cosine series expansion method for pricing financial options," CPB Discussion Paper 225, CPB Netherlands Bureau for Economic Policy Analysis.
    12. Shimizu, Yasutaka, 2009. "A new aspect of a risk process and its statistical inference," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 70-77, February.
    13. You, Honglong & Guo, Junyi & Jiang, Jiancheng, 2020. "Interval estimation of the ruin probability in the classical compound Poisson risk model," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    14. 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.
    15. Comte, F. & Genon-Catalot, V., 2009. "Nonparametric estimation for pure jump Lévy processes based on high frequency data," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4088-4123, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xie, Jiayi & Zhang, Zhimin, 2020. "Statistical estimation for some dividend problems under the compound Poisson risk model," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 101-115.
    2. 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.
    3. Yuan Gao & Honglong You, 2021. "The Speed of Convergence of the Threshold Estimator of Ruin Probability under the Tempered α -Stable Lévy Subordinator," Mathematics, MDPI, vol. 9(21), pages 1-9, October.
    4. Chongkai Xie & Honglong You, 2024. "A Threshold Estimator for Ruin Probability Using the Fourier-Cosine Method in the Wiener–Poisson Risk Model," Mathematics, MDPI, vol. 12(18), pages 1-14, September.
    5. Kang Hu & Ya Huang & Yingchun Deng, 2023. "Estimating the Gerber–Shiu Function in the Two-Sided Jumps Risk Model by Laguerre Series Expansion," Mathematics, MDPI, vol. 11(9), pages 1-30, April.
    6. Yunyun Wang & Wenguang Yu & Yujuan Huang & Xinliang Yu & Hongli Fan, 2019. "Estimating the Expected Discounted Penalty Function in a Compound Poisson Insurance Risk Model with Mixed Premium Income," Mathematics, MDPI, vol. 7(3), pages 1-25, March.
    7. 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.
    8. 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.
    9. Wen Su & Wenguang Yu, 2020. "Asymptotically Normal Estimators of the Gerber-Shiu Function in Classical Insurance Risk Model," Mathematics, MDPI, vol. 8(10), pages 1-11, September.
    10. Shimizu, Yasutaka & Zhang, Zhimin, 2017. "Estimating Gerber–Shiu functions from discretely observed Lévy driven surplus," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 84-98.
    11. Chau, K.W. & Yam, S.C.P. & Yang, H., 2015. "Fourier-cosine method for Gerber–Shiu functions," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 170-180.
    12. Wenguang Yu & Yaodi Yong & Guofeng Guan & Yujuan Huang & Wen Su & Chaoran Cui, 2019. "Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion," Mathematics, MDPI, vol. 7(9), pages 1-15, September.
    13. Zhang, Zhimin & Yang, Hailiang, 2014. "Nonparametric estimation for the ruin probability in a Lévy risk model under low-frequency observation," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 168-177.
    14. Zan Yu & Lianzeng Zhang, 2024. "Computing the Gerber-Shiu function with interest and a constant dividend barrier by physics-informed neural networks," Papers 2401.04378, arXiv.org.
    15. Rong Du & Duy-Minh Dang, 2023. "Fourier Neural Network Approximation of Transition Densities in Finance," Papers 2309.03966, arXiv.org, revised Sep 2024.
    16. Zhimin Zhang & Hailiang Yang & Hu Yang, 2012. "On a Sparre Andersen Risk Model with Time-Dependent Claim Sizes and Jump-Diffusion Perturbation," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 973-995, December.
    17. Muroi, Yoshifumi & Suda, Shintaro, 2022. "Binomial tree method for option pricing: Discrete cosine transform approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 198(C), pages 312-331.
    18. Christian Bayer & Chiheb Ben Hammouda & Antonis Papapantoleon & Michael Samet & Ra'ul Tempone, 2024. "Quasi-Monte Carlo for Efficient Fourier Pricing of Multi-Asset Options," Papers 2403.02832, arXiv.org.
    19. Andersson, Kristoffer & Oosterlee, Cornelis W., 2021. "A deep learning approach for computations of exposure profiles for high-dimensional Bermudan options," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    20. Teng, Ye & Zhang, Zhimin, 2023. "Finite-time expected present value of operating costs until ruin in a Cox risk model with periodic observation," Applied Mathematics and Computation, Elsevier, vol. 452(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:12:p:1402-:d:576424. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.