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

  • Matteo Basei
  • Huyên PhamEmail author
Article
  • 49 Downloads

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.

Keywords

Mean-field SDEs Linear-quadratic optimal control Weak martingale optimality principle Riccati equation 

Mathematics Subject Classification

49N10 49L20 93E20 

Notes

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Industrial Engineering and Operations Research Department (IEOR)University of California, BerkeleyBerkeleyUSA
  2. 2.Laboratoire de Probabilités, Statistique et Modélisation (LPSM)Université Paris Diderot and CREST-ENSAEParisFrance

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