On Time-Consistent Solution to Time-Inconsistent Linear-Quadratic Optimal Control of Discrete-Time Stochastic Systems

TL;DR

This paper analyzes time-inconsistent stochastic linear-quadratic control problems in discrete time, establishing conditions for open-loop and feedback equilibrium solutions via generalized Riccati equations.

Contribution

It provides necessary and sufficient conditions for the existence of time-consistent solutions, linking them to solvability of generalized Riccati equations.

Findings

Existence of open-loop equilibrium control characterized by nonsymmetric Riccati equations.

Existence of feedback equilibrium strategy characterized by symmetric Riccati equations.

Conditions derived are applicable when system matrices are time-independent.

Abstract

In this paper, we investigate a class of time-inconsistent discrete-time stochastic linear-quadratic optimal control problems, whose time-consistent solutions consist of an open-loop equilibrium control and a linear feedback equilibrium strategy. The open-loop equilibrium control is defined for a given initial pair, while the linear feedback equilibrium strategy is defined for all the initial pairs. Maximum-principle-type necessary and sufficient conditions containing stationary and convexity are derived for the existence of these two time-consistent solutions, respectively. Furthermore, for the case where the system matrices are independent of the initial time, we show that the existence of the open-loop equilibrium control for a given initial pair is equivalent to the solvability of a set of nonsymmetric generalized difference Riccati equations and a set of linear difference equations. Moreover, the existence of linear feedback equilibrium strategy is equivalent to the solvability of another set of symmetric generalized difference Riccati equations.