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A Weak Martingale Approach to Linear-Quadratic McKean–Vlasov Stochastic Control Problems

Author

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  • Matteo Basei

    (University of California, Berkeley)

  • Huyên Pham

    (Université Paris Diderot and CREST-ENSAE)

Abstract

We propose a simple and direct approach for solving linear-quadratic mean-field stochastic control problems. We study both finite-horizon and infinite-horizon problems and allow notably some coefficients to be stochastic. Extension to the common noise case is also addressed. Our method is based on a suitable version of the martingale formulation for verification theorems in control theory. The optimal control involves the solution to a system of Riccati ordinary differential equations and to a linear mean-field backward stochastic differential equation; existence and uniqueness conditions are provided for such a system. Finally, we illustrate our results through an application to the production of an exhaustible resource.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:2:d:10.1007_s10957-018-01453-z
    DOI: 10.1007/s10957-018-01453-z
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    References listed on IDEAS

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    1. A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
    2. Olivier Guéant & Pierre Louis Lions & Jean-Michel Lasry, 2011. "Mean Field Games and Applications," Post-Print hal-01393103, HAL.
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    Cited by:

    1. Ren'e Aid & Ofelia Bonesini & Giorgia Callegaro & Luciano Campi, 2021. "A McKean-Vlasov game of commodity production, consumption and trading," Papers 2111.04391, arXiv.org.
    2. René Aïd & Matteo Basei & Huyên Pham, 2020. "A McKean–Vlasov approach to distributed electricity generation development," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(2), pages 269-310, April.
    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. Willliam Lefebvre & Gregoire Loeper & Huyên Pham, 2020. "Mean-variance portfolio selection with tracking error penalization," Working Papers hal-02941289, HAL.
    5. Christoph Belak & Daniel Hoffmann & Frank T. Seifried, 2020. "Continuous-Time Mean Field Games with Finite StateSpace and Common Noise," Working Paper Series 2020-05, University of Trier, Research Group Quantitative Finance and Risk Analysis.
    6. Maximilien Germain & Mathieu Laurière & Huyên Pham & Xavier Warin, 2022. "DeepSets and their derivative networks for solving symmetric PDEs ," Post-Print hal-03154116, HAL.

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