Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I

TL;DR

This paper studies time inconsistent mean-field stochastic linear quadratic control problems, introducing equilibrium controls, characterizing them via Riccati equations, and discussing their uniqueness within a Markovian framework.

Contribution

It develops new characterizations of equilibrium controls for mean-field LQ problems with time inconsistency, including coupled Riccati equations and novel definitions.

Findings

Derived coupled Riccati equations for equilibrium controls.

Characterized open-loop and closed-loop equilibrium controls.

Proved uniqueness of open-loop equilibrium controls.

Abstract

In this paper, a class of time inconsistent linear quadratic optimal control problems of mean-field stochastic differential equations (SDEs) is considered under Markovian framework. Open-loop equilibrium controls and their particular closed-loop representations are introduced and characterized via variational ideas. Several interesting features are revealed and a system of coupled Riccati equations is derived. In contrast with the analogue optimal control problems of SDEs, the mean-field terms in state equation, which is another reason of time inconsistency, prompts us to define above two notions in new manners. An interesting result, which is almost trivial in the counterpart problems of SDEs, is given and plays significant role in the previous characterizations. As application, the uniqueness of open-loop equilibrium controls is discussed.