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

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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.

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Notes

  1. We thank René Aid for insightful discussions on this example.

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Acknowledgements

This work is part of the ANR Project CAESARS (ANR-15-CE05-0024) and also supported by FiME (Finance for Energy Market Research Centre) and the “Finance et Développement Durable—Approches Quantitatives” EDF—CACIB Chair.

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Authors and Affiliations

  1. Industrial Engineering and Operations Research Department (IEOR), University of California, Berkeley, Berkeley, USA

    Matteo Basei

  2. Laboratoire de Probabilités, Statistique et Modélisation (LPSM), Université Paris Diderot and CREST-ENSAE, Paris, France

    Huyên Pham

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  1. Matteo Basei
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  2. Huyên Pham
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Correspondence to Huyên Pham.

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Communicated by Bruno Bouchard.

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Basei, M., Pham, H. A Weak Martingale Approach to Linear-Quadratic McKean–Vlasov Stochastic Control Problems. J Optim Theory Appl 181, 347–382 (2019). https://doi.org/10.1007/s10957-018-01453-z

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