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On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes

Author

Listed:
  • Florin Avram

    (Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France)

  • Andras Horváth

    (Dipartimento di Informatica, Università di Torino, Corso Svizzera 185, 10149 Torino, Italy)

  • Serge Provost

    (Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, ON N6A5B7, Canada)

  • Ulyses Solon

    (Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France)

Abstract

This paper considers the Brownian perturbed Cramér–Lundberg risk model with a dividends barrier. We study various types of Padé approximations and Laguerre expansions to compute or approximate the scale function that is necessary to optimize the dividends barrier. We experiment also with a heavy-tailed claim distribution for which we apply the so-called “shifted” Padé approximation.

Suggested Citation

  • Florin Avram & Andras Horváth & Serge Provost & Ulyses Solon, 2019. "On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes," Risks, MDPI, vol. 7(4), pages 1-24, December.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:4:p:121-:d:296915
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    References listed on IDEAS

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    1. Pablo Azcue & Nora Muler, 2005. "Optimal Reinsurance And Dividend Distribution Policies In The Cramér‐Lundberg Model," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 261-308, April.
    2. Gerber, Hans U. & Shiu, Elias S.W. & Smith, Nathaniel, 2008. "Methods for estimating the optimal dividend barrier and the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 243-254, February.
    3. Merton H. Miller & Franco Modigliani, 1961. "Dividend Policy, Growth, and the Valuation of Shares," The Journal of Business, University of Chicago Press, vol. 34, pages 411-411.
    4. Bingham, N. H., 1976. "Continuous branching processes and spectral positivity," Stochastic Processes and their Applications, Elsevier, vol. 4(3), pages 217-242, August.
    5. Florin Avram & Zbigniew Palmowski & Martijn R. Pistorius, 2007. "On the optimal dividend problem for a spectrally negative L\'{e}vy process," Papers math/0702893, arXiv.org.
    6. Hu, Xiang & Duan, Baige & Zhang, Lianzeng, 2017. "De Vylder approximation to the optimal retention for a combination of quota-share and excess of loss reinsurance with partial information," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 48-55.
    7. Avram, F. & Pistorius, M., 2014. "On matrix exponential approximations of ruin probabilities for the classic and Brownian perturbed Cramér–Lundberg processes," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 57-64.
    8. F. Avram & Z. Palmowski & M. R. Pistorius, 2011. "On Gerber-Shiu functions and optimal dividend distribution for a L\'{e}vy risk process in the presence of a penalty function," Papers 1110.4965, arXiv.org, revised Jun 2015.
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    Cited by:

    1. Florin Avram & Dan Goreac & Rim Adenane & Ulyses Solon, 2022. "Optimizing Dividends and Capital Injections Limited by Bankruptcy, and Practical Approximations for the Cramér-Lundberg Process," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2339-2371, December.
    2. Albrecher, Hansjörg & Cheung, Eric C.K. & Liu, Haibo & Woo, Jae-Kyung, 2022. "A bivariate Laguerre expansions approach for joint ruin probabilities in a two-dimensional insurance risk process," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 96-118.

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