Time-Inconsistent Mean-Field Stochastic LQ Problem: Open-Loop Time-Consistent Control

TL;DR

This paper develops a framework for finding open-loop time-consistent equilibrium controls in time-inconsistent mean-field stochastic LQ problems, using difference equations and Riccati equations.

Contribution

It introduces necessary and sufficient conditions for the existence of such controls, extending classical LQ control theory to time-inconsistent mean-field settings.

Findings

Equivalence between equilibrium control existence and solvability of difference equations.

Decoupling of forward-backward stochastic difference equations.

Characterization of controls via generalized difference Riccati equations.

Abstract

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.