# American Institute of Mathematical Sciences

June  2019, 9(2): 385-409. doi: 10.3934/mcrf.2019018

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

 School of Mathematics, Sichuan University, Chengdu, China

Received  November 2017 Revised  August 2018 Published  December 2018

Fund Project: The research was supported by the NSF of China under grant 11231007, 11401404 and 11471231, and the Fundamental Research Funds for the central Universities (YJ201605).

In this paper, a class of time inconsistent linear quadratic optimal control problems for mean-field stochastic differential equations (SDEs) are 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 the 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.

Citation: Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018
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