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Optimality of a refraction strategy in the optimal dividends problem with absolutely continuous controls subject to Parisian ruin

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  • Locas, Félix
  • Renaud, Jean-François

Abstract

We consider de Finetti's optimal dividends problem with absolutely continuous strategies in a spectrally negative Lévy model with Parisian ruin as the termination time. The problem considered is essentially a generalization of both the control problems considered by Kyprianou et al. (2012) and by Renaud (2019). Using the language of scale functions for Parisian fluctuation theory, and under the assumption that the density of the Lévy measure is completely monotone, we prove that a refraction dividend strategy is optimal and we characterize the optimal threshold. In particular, we study the effect of the rate of Parisian implementation delays on this optimal threshold.

Suggested Citation

  • Locas, Félix & Renaud, Jean-François, 2025. "Optimality of a refraction strategy in the optimal dividends problem with absolutely continuous controls subject to Parisian ruin," Insurance: Mathematics and Economics, Elsevier, vol. 120(C), pages 189-206.
    • Handle: RePEc:eee:insuma:v:120:y:2025:i:c:p:189-206
    DOI: 10.1016/j.insmatheco.2024.11.011
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    References listed on IDEAS

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    1. Jean-François Renaud, 2019. "De Finetti’s Control Problem with Parisian Ruin for Spectrally Negative Lévy Processes," Risks, MDPI, vol. 7(3), pages 1-11, July.
    2. Irmina Czarna & Zbigniew Palmowski, 2014. "Dividend Problem with Parisian Delay for a Spectrally Negative Lévy Risk Process," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 239-256, April.
    3. 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.
    4. Loeffen, Ronnie L. & Renaud, Jean-François, 2010. "De Finetti's optimal dividends problem with an affine penalty function at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 98-108, February.
    5. Andreas E. Kyprianou & Víctor Rivero & Renming Song, 2010. "Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem," Journal of Theoretical Probability, Springer, vol. 23(2), pages 547-564, June.
    6. Mohamed Amine Lkabous & Jean-François Renaud, 2019. "A unified approach to ruin probabilities with delays for spectrally negative Lévy processes," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2019(8), pages 711-728, September.
    7. Hansjörg Albrecher & Eric Cheung & Stefan Thonhauser, 2013. "Randomized observation periods for the compound Poisson risk model: the discounted penalty function," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2013(6), pages 424-452.
    8. Landriault, David & Renaud, Jean-François & Zhou, Xiaowen, 2011. "Occupation times of spectrally negative Lévy processes with applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2629-2641, November.
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    10. Renaud, Jean-François, 2024. "A note on the optimal dividends problem with transaction costs in a spectrally negative Lévy model with Parisian ruin," Statistics & Probability Letters, Elsevier, vol. 206(C).
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