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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 problems, and allow notably some coefficients to bestochastic. Our method is based on a suitable extension of the martingaleformulation for verification theorems in control theory. The optimal controlinvolves the solution to a system of Riccati ordinary differential equationsand to a linear mean-field backward stochastic differential equation, existenceand uniqueness conditions are provided for such a system. Finally, weillustrate our results through two applications with explicit solutions: thefirst one deals with a portfolio liquidation problem with trade crowding, andthe second one considers an economic model of substitutable production goods.

Linear-quadratic McKean-Vlasov stochastic control problems with random coefficients on finite and infinite horizon, and applications