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McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations

Author

Listed:
  • Mao Fabrice Djete

    (Centre de Mathématiques Appliquées, École Polytechnique, 91120 Palaiseau, France)

  • Dylan Possamaï

    (Department of Mathematics, Eidgenössische Technische Hochschule Zürich, 8092 Zürich, Switzerland)

  • Xiaolu Tan

    (Department of Mathematics, The Chinese University of Hong Kong, Hong Kong)

Abstract

We study a McKean–Vlasov optimal control problem with common noise in order to establish the corresponding limit theory as well as the equivalence between different formulations, including strong, weak, and relaxed formulations. In contrast to the strong formulation, in which the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an ( H )-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classic optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean–Vlasov control problem, that is, proving that it consists in the limit of a large population control problem with common noise.

Suggested Citation

  • Mao Fabrice Djete & Dylan Possamaï & Xiaolu Tan, 2022. "McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations," Mathematics of Operations Research, INFORMS, vol. 47(4), pages 2891-2930, November.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:4:p:2891-2930
    DOI: 10.1287/moor.2021.1232
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    References listed on IDEAS

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    2. Romuald Élie & Emma Hubert & Thibaut Mastrolia & Dylan Possamaï, 2021. "Mean–field moral hazard for optimal energy demand response management," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 399-473, January.
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    5. Romuald Elie & Thibaut Mastrolia & Dylan Possamaï, 2019. "A Tale of a Principal and Many, Many Agents," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 440-467, May.
    6. Matteo Basei & Huyên Pham, 2019. "A Weak Martingale Approach to Linear-Quadratic McKean–Vlasov Stochastic Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 347-382, May.
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    Cited by:

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    7. Felix Hofer, 2025. "Stationary Heterogeneous-Agent Models in Continuous Time," Papers 2510.26065, arXiv.org, revised Jun 2026.

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