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On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums

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  • Ewa Marciniak

    (AGH University of Science and Technology)

  • Zbigniew Palmowski

    (University of Wrocław)

Abstract

This paper concerns an optimal dividend distribution problem for an insurance company with surplus-dependent premium. In the absence of dividend payments, such a risk process is a particular case of so-called piecewise deterministic Markov processes. The control mechanism chooses the size of dividend payments. The objective consists in maximizing the sum of the expected cumulative discounted dividend payments received until the time of ruin and a penalty payment at the time of ruin, which is an increasing function of the size of the shortfall at ruin. A complete solution is presented to the corresponding stochastic control problem. We identify the associated Hamilton–Jacobi–Bellman equation and find necessary and sufficient conditions for optimality of a single dividend-band strategy, in terms of particular Gerber–Shiu functions. A number of concrete examples are analyzed.

Suggested Citation

  • Ewa Marciniak & Zbigniew Palmowski, 2016. "On the Optimal Dividend Problem for Insurance Risk Models with Surplus-Dependent Premiums," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 723-742, February.
    • Handle: RePEc:spr:joptap:v:168:y:2016:i:2:d:10.1007_s10957-015-0755-3
    DOI: 10.1007/s10957-015-0755-3
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    References listed on IDEAS

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    1. Avanzi, Benjamin & Gerber, Hans U., 2008. "Optimal Dividends in the Dual Model with Diffusion," ASTIN Bulletin, Cambridge University Press, vol. 38(2), pages 653-667, November.
    2. 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.
    3. Gerber, Hans U. & Smith, Nathaniel, 2008. "Optimal dividends with incomplete information in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 227-233, October.
    4. Bjarne Højgaard & Søren Asmussen & Michael Taksar, 2000. "Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation," Finance and Stochastics, Springer, vol. 4(3), pages 299-324.
    5. 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.
    6. 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.
    7. Loeffen, R.L., 2009. "An optimal dividends problem with transaction costs for spectrally negative Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 41-48, August.
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    Cited by:

    1. Ewa Marciniak & Zbigniew Palmowski, 2018. "On the Optimal Dividend Problem in the Dual Model with Surplus-Dependent Premiums," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 533-552, November.
    2. Osatakul, Dhiti & Li, Shuanming & Wu, Xueyuan, 2023. "Discrete-time risk models with surplus-dependent premium corrections," Applied Mathematics and Computation, Elsevier, vol. 437(C).
    3. Ewa Marciniak & Zbigniew Palmowski, 2016. "On the Optimal Dividend Problem in the Dual Model with Surplus-Dependent Premiums," Papers 1605.04584, arXiv.org.
    4. Gordienko, E. & Vázquez-Ortega, P., 2018. "Continuity inequalities for multidimensional renewal risk models," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 48-54.
    5. Florin Avram & Jose-Luis Perez-Garmendia, 2019. "A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems," Risks, MDPI, vol. 7(4), pages 1-21, November.
    6. Linlin Tian & Lihua Bai & Junyi Guo, 2020. "Optimal Singular Dividend Problem Under the Sparre Andersen Model," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 603-626, February.

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