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Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates

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  • Azcue, Pablo
  • Muler, Nora

Abstract

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cramér–Lundberg model with arbitrary claim-size distribution. Our objective is to find the dividend payment policy which maximizes the cumulative expected discounted dividend pay-outs until the time of bankruptcy imposing a ceiling on the dividend rates. We characterize the optimal value function as the unique bounded viscosity solution of the associated Hamilton–Jacobi–Bellman equation. We prove that there exists an optimal dividend strategy and that this strategy is stationary with a band structure. We study the regularity of the optimal value function. We find a characterization result to check optimality even in the case where the optimal value function is not differentiable. We construct examples where the claim-size distribution is smooth but the optimal dividend policy is not threshold and the optimal value function is not differentiable. We study the survival probability of the company under the optimal dividend policy. We also present examples where the optimal dividend policy has infinitely many bands even in the case that the claim-size distribution has a bounded density.

Suggested Citation

  • Azcue, Pablo & Muler, Nora, 2012. "Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 26-42.
    • Handle: RePEc:eee:insuma:v:51:y:2012:i:1:p:26-42
    DOI: 10.1016/j.insmatheco.2012.02.011
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    References listed on IDEAS

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    1. Benjamin Avanzi, 2009. "Strategies for Dividend Distribution: A Review," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(2), pages 217-251.
    2. Kristin Reikvam & Fred Espen Benth & Kenneth Hvistendahl Karlsen, 2001. "Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach," Finance and Stochastics, Springer, vol. 5(3), pages 275-303.
    3. Hans Gerber & Elias Shiu, 2006. "On Optimal Dividend Strategies In The Compound Poisson Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(2), pages 76-93.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Philipp Lukas Strietzel & Henriette Elisabeth Heinrich, 2022. "Optimal Dividends for a Two-Dimensional Risk Model with Simultaneous Ruin of Both Branches," Risks, MDPI, vol. 10(6), pages 1-23, June.
    2. Feng, Yang & Siu, Tak Kuen & Zhu, Jinxia, 2024. "Optimal payout strategies when Bruno de Finetti meets model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 148-164.
    3. Chen, Shumin & Li, Zhongfei & Zeng, Yan, 2014. "Optimal dividend strategies with time-inconsistent preferences," Journal of Economic Dynamics and Control, Elsevier, vol. 46(C), pages 150-172.
    4. Linlin Tian & Xiaoyi Zhang, 2018. "Optimal Dividend of Compound Poisson Process under a Stochastic Interest Rate," Papers 1807.08081, arXiv.org.
    5. Hansjoerg Albrecher & Pablo Azcue & Nora Muler, 2019. "Optimal ratcheting of dividends in insurance," Papers 1910.06910, arXiv.org, revised Jun 2021.
    6. Xu, Ran & Woo, Jae-Kyung, 2020. "Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 1-16.
    7. Yao, Dingjun & Yang, Hailiang & Wang, Rongming, 2014. "Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle," Economic Modelling, Elsevier, vol. 37(C), pages 53-64.

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    More about this item

    Keywords

    Cramér–Lundberg process; Insurance; Bounded dividend rates; Optimal investment policy; Hamilton–Jacobi–Bellman equation; Viscosity solution; Risk control; Threshold strategy; Band strategy;
    All these keywords.

    JEL classification:

    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
    • G35 - Financial Economics - - Corporate Finance and Governance - - - Payout Policy

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