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We propose a simple and original approach for solving linear-quadraticmean-field stochastic control problems. We study both finite-horizon andinfinite-horizon pro\-blems, and allow notably some coefficients to bestochastic. Extension to the common noise case is also addressed. Our method isbased on a suitable version of the martingale formulation for verificationtheorems in control theory. The optimal control involves the solution to asystem of Riccati ordinary differential equations and to a linear mean-fieldbackward stochastic differential equation; existence and uniqueness conditionsare provided for such a system. Finally, we illustrate our results through anapplication to the production of an exhaustible resource. MSC Classification:49N10, 49L20, 93E20.

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