  This paper is concerned with the open-loop time-consistent solution of
time-inconsistent mean-field stochastic linear-quadratic optimal control.
Different from standard stochastic linear-quadratic problems, both the system
matrices and the weighting matrices are dependent on the initial times, and the
conditional expectations of the control and state enter quadratically into the
cost functional. Such features will ruin Bellman's principle of optimality and
result in the time-inconsistency of the optimal control. Based on the dynamical
nature of the systems involved, a kind of open-loop time-consistent equilibrium
control is investigated in this paper. It is shown that the existence of
open-loop time-consistent equilibrium control for a fixed initial pair is
equivalent to the solvability of a set of forward-backward stochastic
difference equations with stationary conditions and convexity conditions. By
decoupling the forward-backward stochastic difference equations, necessary and
sufficient conditions in terms of linear difference equations and generalized
difference Riccati equations are given for the existence of open-loop
time-consistent equilibrium control with a fixed initial pair. Moreover, the
existence of open-loop time-consistent equilibrium control for all the initial
pairs is shown to be equivalent to the solvability of a set of coupled
constrained generalized difference Riccati equations and two sets of
constrained linear difference equations.
