Linear-quadratic optimal control problems for mean-field stochastic differential equations — time-consistent solutions

Linear-quadratic optimal control problems are considered for mean-field stochastic differential equations with deterministic coefficients. Time-inconsistency feature of the problems is carefully investigated. Both open-loop and closed-loop equilibrium solutions are presented for such kinds of problems. Open-loop solutions are presented by means of the variational method with decoupling of forward-backward stochastic differential equations, which leads to a Riccati equation system lack of symmetry. Closed-loop solutions are presented by means of multi-person differential games, the limit of which leads to a Riccati equation system with a symmetric structure.