Time-inconsistent mean-field stochastic linear-quadratic optimal control

This paper is concerned with a kind of time-consistent solution of the 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 initial time, and the conditional expectations of the control and state enter quadratically into the cost functional. Such features will ruin the Bellman optimality principle and result in the time-inconsistency of the optimal control. Due to the dynamical nature of control problems, a kind of open-loop time-consistent equilibrium control is thoroughly investigated in this paper. It is shown that the existence of an 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, the existence of the open-loop equilibrium control for all the initial pairs is shown to be equivalent to the solvability of a set of coupled constraint generalized difference Riccati equations and a set of coupled constraint linear difference equations.