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Linearisation Techniques and the Dual Algorithm for a Class of Mixed Singular/Continuous Control Problems in Reinsurance. Part I: Theoretical Aspects

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  • GOREAC, Dan
  • LI, Juan
  • XU, Boxiang

Abstract

This paper focuses on linearisation techniques for a class of mixed singular/continuous control problems and ensuing algorithms. The motivation comes from (re)insurance problems with reserve-dependent premiums with Cramér-Lundberg claims, by allowing singular dividend payments and capital injections. Using variational techniques and embedding the trajectories in an appropriate family of occupation measures, we provide the linearisation of such problems in which the continuous control is given by reinsurance policies and the singular one by dividends and capital injections. The linearisation translates into a dual dynamic programming (DDP) algorithm. An important part of the paper is dedicated to structural considerations allowing reasonable implementation. We also hint connections to methods relying on moment sum of squares and LMI (linear matrix inequality)-relaxations to approximate the optimal candidates.

Suggested Citation

  • GOREAC, Dan & LI, Juan & XU, Boxiang, 2022. "Linearisation Techniques and the Dual Algorithm for a Class of Mixed Singular/Continuous Control Problems in Reinsurance. Part I: Theoretical Aspects," Applied Mathematics and Computation, Elsevier, vol. 431(C).
    • Handle: RePEc:eee:apmaco:v:431:y:2022:i:c:s0096300322003952
    DOI: 10.1016/j.amc.2022.127321
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    References listed on IDEAS

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    1. Florin Avram & Dan Goreac & Jean-François Renaud, 2019. "The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps," Risks, MDPI, vol. 7(4), pages 1-9, December.
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    4. Hohmann, Marc & Warrington, Joseph & Lygeros, John, 2020. "A moment and sum-of-squares extension of dual dynamic programming with application to nonlinear energy storage problems," European Journal of Operational Research, Elsevier, vol. 283(1), pages 16-32.
    5. Florin Avram & Dan Goreac & Juan Li & Xiaochi Wu, 2021. "Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model," Mathematics, MDPI, vol. 9(9), pages 1-27, April.
    6. Shapiro, Alexander, 2011. "Analysis of stochastic dual dynamic programming method," European Journal of Operational Research, Elsevier, vol. 209(1), pages 63-72, February.
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    8. Kulenko, Natalie & Schmidli, Hanspeter, 2008. "Optimal dividend strategies in a Cramér-Lundberg model with capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 270-278, October.
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    1. Goreac, Dan & Li, Juan & Wang, Pangbo & Xu, Boxiang, 2024. "Linearisation techniques and the dual algorithm for a class of mixed singular/continuous control problems in reinsurance. Part II: Numerical aspects," Applied Mathematics and Computation, Elsevier, vol. 473(C).

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