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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

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  • Maximilien Germain

    (EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie - EDF R&D - EDF R&D - EDF - EDF, EDF R&D - EDF R&D - EDF - EDF, EDF - EDF, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

  • Huyên Pham

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité, CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique, FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CREST - EDF R&D - EDF R&D - EDF - EDF)

  • Xavier Warin

    (EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie - EDF R&D - EDF R&D - EDF - EDF, EDF R&D - EDF R&D - EDF - EDF, EDF - EDF, FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CREST - EDF R&D - EDF R&D - EDF - EDF)

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, Laurière, 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 & Huyên Pham & Xavier Warin, 2022. "A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection," Post-Print hal-03498263, HAL.
    • Handle: RePEc:hal:journl:hal-03498263
    Note: View the original document on HAL open archive server: https://hal.science/hal-03498263v2
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    References listed on IDEAS

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    1. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2013. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Sciences Po publications info:hdl:2441/5rkqqmvrn4t, Sciences Po.
    2. Willliam Lefebvre & Gregoire Loeper & Huyên Pham, 2020. "Mean-variance portfolio selection with tracking error penalization," Working Papers hal-02941289, HAL.
    3. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Post-Print hal-03460952, HAL.
    4. Abebe Geletu & Michael Klöppel & Hui Zhang & Pu Li, 2013. "Advances and applications of chance-constrained approaches to systems optimisation under uncertainty," International Journal of Systems Science, Taylor & Francis Journals, vol. 44(7), pages 1209-1232.
    5. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," SciencePo Working papers hal-03460952, HAL.
    6. Yuk-Loong Chow & Xiang Yu & Chao Zhou, 2020. "On Dynamic Programming Principle for Stochastic Control Under Expectation Constraints," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 803-818, June.
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    Keywords

    mean-field control; state constraints; neural networks;
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