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Optimal Dividends for a Two-Dimensional Risk Model with Simultaneous Ruin of Both Branches

Author

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  • Philipp Lukas Strietzel

    (Institut für Mathematische Stochastik, Technische Universität Dresden, 01062 Dresden, Germany
    These authors contributed equally to this work.)

  • Henriette Elisabeth Heinrich

    (Institut für Mathematische Stochastik, Technische Universität Dresden, 01062 Dresden, Germany
    These authors contributed equally to this work.)

Abstract

We consider the optimal dividend problem in the so-called degenerate bivariate risk model under the assumption that the surplus of one branch may become negative. More specific, we solve the stochastic control problem of maximizing discounted dividends until simultaneous ruin of both branches of an insurance company by showing that the optimal value function satisfies a certain Hamilton–Jacobi–Bellman (HJB) equation. Further, we prove that the optimal value function is the smallest viscosity solution of said HJB equation, satisfying certain growth conditions. Under some additional assumptions, we show that the optimal strategy lies within a certain subclass of all admissible strategies and reduce the two-dimensional control problem to a one-dimensional one. The results are illustrated by a numerical example and Monte Carlo simulated value functions.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jrisks:v:10:y:2022:i:6:p:116-:d:830383
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    References listed on IDEAS

    as
    1. Thonhauser, Stefan & Albrecher, Hansjorg, 2007. "Dividend maximization under consideration of the time value of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 163-184, July.
    2. Irmina Czarna & Zbigniew Palmowski, 2009. "De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process," Papers 0906.2100, arXiv.org, revised Feb 2011.
    3. Benjamin Avanzi, 2009. "Strategies for Dividend Distribution: A Review," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(2), pages 217-251.
    4. 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.
    5. Pablo Azcue & Nora Muler, 2010. "Optimal investment policy and dividend payment strategy in an insurance company," Papers 1010.4988, arXiv.org.
    6. Pablo Azcue & Nora Muler & Zbigniew Palmowski, 2016. "Optimal dividend payments for a two-dimensional insurance risk process," Papers 1603.07019, arXiv.org, revised Apr 2018.
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