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

Maximilien Germain (), Huyên Pham and Xavier Warin
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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 - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - IP Paris - Institut Polytechnique de Paris - 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

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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.

Keywords: mean-field control; state constraints; neural networks (search for similar items in EconPapers)
Date: 2022
New Economics Papers: this item is included in nep-big, nep-cmp and nep-ene
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Published in Numerical Algebra, Control and Optimization, inPress

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