Finite-Horizon Indefinite Mean-Field Stochastic Linear-Quadratic Optimal Control

For the finite-horizon indefinite mean-field stochastic linear-quadratic optimal control problems, the open-loop optimal control and the closed-loop optimal strategy are introduced and investigated together with their characterizations, difference and relationship. The open-loop optimal control can be defined for a fixed initial state, whose existence is characterized via the solvability of a linear mean-field forward-backward stochastic difference equations with stationary conditions. Differently, the closed-loop strategy is a global notion, which involves all the initial pairs. The existence of the closed-loop optimal strategy is shown to be equivalent to the solvability of a couple of generalized difference Riccati equations, the finiteness of the value function for all the initial pairs, and the existence of open-loop optimal strategy for all the initial pairs.