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This paper first presents necessary and sufficient conditions for thesolvability of discrete time, mean-field, stochastic linear-quadratic optimalcontrol problems. Then, by introducing several sequences of bounded linearoperators, the problem becomes an operator stochastic LQ problem, in which theoptimal control is a linear state feedback. Furthermore, from the form of theoptimal control, the problem changes to a matrix dynamic optimization problem.Solving this optimization problem, we obtain the optimal feedback gain and thusthe optimal control. Finally, by completing the square, the optimality of theabove control is validated.

Discrete Time Mean-Field Stochastic Linear-Quadratic Optimal Control Problems