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A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection

Author

Listed:
  • Maximilien Germain

    (EDF R\&D OSIRIS, EDF R\&D, EDF, LPSM)

  • Huy^en Pham

    (LPSM)

  • Xavier Warin

    (EDF R\&D OSIRIS, EDF R\&D, EDF, FiME Lab)

Abstract

We consider the control of McKean-Vlasov dynamics (or mean-field control) with probabilistic state constraints. We rely on a level-set approach which provides a representation of the constrained problem in terms of an unconstrained one with exact penalization and running maximum or integral cost. The method is then extended to the common noise setting. Our work extends (Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016), pp. 2568--2593) and (Bokanowski, Picarelli, and Zidani, Appl. Math. Optim. 71 (2015), pp. 125--163) to a mean-field setting. The reformulation as an unconstrained problem is particularly suitable for the numerical resolution of the problem, that is achieved from an extension of a machine learning algorithm from (Carmona, Lauri{\`e}re, arXiv:1908.01613 to appear in Ann. Appl. Prob., 2019). A first application concerns the storage of renewable electricity in the presence of mean-field price impact and another one focuses on a mean-variance portfolio selection problem with probabilistic constraints on the wealth. We also illustrate our approach for a direct numerical resolution of the primal Markowitz continuous-time problem without relying on duality.

Suggested Citation

  • Maximilien Germain & Huy^en Pham & Xavier Warin, 2021. "A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection," Papers 2112.11059, arXiv.org, revised Nov 2022.
    • Handle: RePEc:arx:papers:2112.11059
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    References listed on IDEAS

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    1. Bruno Bouchard & Boualem Djehiche & Idris Kharroubi, 2020. "Quenched Mass Transport of Particles Toward a Target," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 345-374, August.
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    3. William Lefebvre & Gregoire Loeper & Huy^en Pham, 2020. "Mean-variance portfolio selection with tracking error penalization," Papers 2009.08214, arXiv.org, revised Sep 2020.
    4. Xavier Warin, 2021. "Reservoir optimization and Machine Learning methods," Papers 2106.08097, arXiv.org, revised May 2023.
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    7. Maximilien Germain & Mathieu Laurière & Huyên Pham & Xavier Warin, 2021. "DeepSets and their derivative networks for solving symmetric PDEs ," Working Papers hal-03154116, HAL.
    8. Silvana Pesenti & Sebastian Jaimungal, 2020. "Portfolio Optimisation within a Wasserstein Ball," Papers 2012.04500, arXiv.org, revised Jun 2022.
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    10. Balata, Alessandro & Ludkovski, Michael & Maheshwari, Aditya & Palczewski, Jan, 2021. "Statistical learning for probability-constrained stochastic optimal control," European Journal of Operational Research, Elsevier, vol. 290(2), pages 640-656.
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