Time-Inconsistent Discrete-Time Stochastic Linear-Quadratic Optimal Control: Time-consistent Solutions

In this paper, the time-consistent solutions of a timeinconsistent discrete-time stochastic linear-quadratic optimal control are investigated. Different from the existing literature, the definiteness constraint is not posed on the state and the control weight matrices of the cost functional. Necessary and sufficient conditions are, respectively, obtained to the existence of the open-loop time-consistent equilibrium control and the closedloop time-consistent equilibrium strategy, which contain the solvability of certain system of forward-backward stochastic difference equations, the stationary conditions and the convexity conditions. Furthermore, for the case that the system matrices are independent of the initial time, the existence of the openloop equilibrium pair for all the initial pairs is shown to be equivalent to the solvability of a system of generalized difference Riccati equations and a system of linear difference equations. Moreover, the existence of the closed-loop equilibrium strategy is shown to be equivalent to the solvability of another system of generalized difference Riccati equations. Interestingly, if solvable, the system of generalized difference Riccati equations for the open-loop formulation does not admit symmetric solution, while the one for the closed-loop formulation has symmetric solution.