Discrete-Time Stochastic Linear-Quadratic Optimal Control with Time-Inconsistency

Abstract

In this paper, the time-consistent solution of a discrete-time time-inconsistent stochastic linear-quadratic optimal control is investigated. Different from existing literature, the definiteness constraint is not posed on the state weight matrices and the control weight matrices of the cost functional. Necessary and suffcient conditions are obtained to the existence of the open-loop time-consistent equilibrium control, which contain the solvability of certain forward-backward stochastic difference equation systems, the stationary conditions and the convexity conditions. Under additional conditions, the closed-form of the open-loop equilibrium control is characterized via the solutions of systems of certain generalized difference Riccati equations. Interestingly, the system of generalized difference Riccati equations do not admit symmetry structure. Finally, for a special case of the considered problem, the existence of the open-loop equilibrium control for all the initial pairs is shown to be equivalent to the solvability of certain generalized difference Riccati equation.