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Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model

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  • Julia Eisenberg

    (Department of Financial and Actuarial Mathematics, TU Wien, Wiedner Hauptstraße 8–10/E105-1, 1040 Vienna, Austria)

  • Lukas Fabrykowski

    (Triangular IT Solutions e.U., 1220 Vienna, Austria)

  • Maren Diane Schmeck

    (Center for Mathematical Economics, Bielefeld University, Universitätsstraße 25, 33615 Bielefeld, Germany)

Abstract

In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and the reinsurance price is modelled by a continuous-time Markov chain with two states. The presence of regime-switching substantially complicates the optimal reinsurance problem, as the surplus-independent strategies turn out to be suboptimal. We develop a recursive approach that allows to represent a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation and the corresponding reinsurance strategy as the unique limits of the sequence of solutions to ordinary differential equations and their first- and second-order derivatives. Via Ito’s formula, we prove the constructed function to be the value function. Two examples illustrate the recursive procedure along with a numerical approach yielding the direct solution to the HJB equation.

Suggested Citation

  • Julia Eisenberg & Lukas Fabrykowski & Maren Diane Schmeck, 2021. "Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model," Risks, MDPI, vol. 9(4), pages 1-25, April.
    • Handle: RePEc:gam:jrisks:v:9:y:2021:i:4:p:73-:d:535490
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    References listed on IDEAS

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    1. Matteo Brachetta & Claudia Ceci, 2019. "A BSDE-based approach for the optimal reinsurance problem under partial information," Papers 1910.05999, arXiv.org, revised May 2020.
    2. Zied Ben Salah & Jos'e Garrido, 2017. "On Fair Reinsurance Premiums; Capital Injections in a Perturbed Risk Model," Papers 1710.11065, arXiv.org, revised Jun 2018.
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    5. Ben Salah, Zied & Garrido, José, 2018. "On fair reinsurance premiums; Capital injections in a perturbed risk model," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 11-20.
    6. Mathieu Bargès & Stéphane Loisel & Xavier Venel, 2011. "On finite-time ruin probabilities with reinsurance cycles influenced by large claims," Post-Print hal-00430178, HAL.
    7. Eisenberg, Julia & Krühner, Paul, 2018. "The impact of negative interest rates on optimal capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 1-10.
    8. Zhengjun Jiang & Martijn Pistorius, 2012. "Optimal dividend distribution under Markov regime switching," Finance and Stochastics, Springer, vol. 16(3), pages 449-476, July.
    9. Pierre‐André Chiappori & Bruno Jullien & Bernard Salanié & François Salanié, 2006. "Asymmetric information in insurance: general testable implications," RAND Journal of Economics, The RAND Corporation, vol. 37(4), pages 783-798, December.
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    Cited by:

    1. Federico, Salvatore & Ferrari, Giorgio & Torrente, Maria Laura, 2023. "Irreversible Reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost," Center for Mathematical Economics Working Papers 682, Center for Mathematical Economics, Bielefeld University.
    2. Abbaspour, Manijeh & Vajargah, Kianoush Fathi & Azhdari, Parvin, 2023. "An efficient algorithm for pricing reinsurance contract under the regime-switching model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 278-300.

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