A Weak Martingale Approach to Linear-Quadratic McKean–Vlasov Stochastic Control Problems
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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.
KeywordsMean-field SDEs Linear-quadratic optimal control Weak martingale optimality principle Riccati equation
Mathematics Subject Classification49N10 49L20 93E20
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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