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

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doi:10.3934/mcrf.2019018

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2019, Volume 9, Issue 2: 385-409. Doi: 10.3934/mcrf.2019018

This issue Previous Article Nonlinear Schrödinger equations on a finite interval with point dissipation Next Article

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: June 2019

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)

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Abstract

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.

Keywords:

Mean-field linear quadratic optimal control problems,
time inconsistency,
equilibrium controls,
system of Riccati equations.

Mathematics Subject Classification: Primary: 49N10, 49N35; Secondary: 93E20.

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