Linear-quadratic-Gaussian mean-field-game with partial observation and common noise

Bensoussan Alain; Feng Xinwei; Huang Jianhui

doi:10.3934/mcrf.2020025

Article Contents

2021, Volume 11, Issue 1: 23-46. Doi: 10.3934/mcrf.2020025

This issue Previous Article Optimal dividend policy in an insurance company with contagious arrivals of claims Next Article Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability

Linear-quadratic-Gaussian mean-field-game with partial observation and common noise

1.
International Center for Decision and Risk Analysis Jindal School of Management, The University of Texas at Dallas and School of Data Sciences, City University of Hong Kong
2.
Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, China
3.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

Received: December 31, 2018

Revised: January 31, 2020

Early access: May 2020

Published: February 2021

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Abstract

This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related $ \epsilon $-Nash equilibrium property is verified.

Keywords:

Mean-field games,
$ \epsilon $-Nash equilibrium,
forward-backward stochastic differential equation,
Kalman filtering,
partial observation,
consistency condition.

Mathematics Subject Classification: Primary: 60H10, 91A10, 91A23; Secondary: 60H30, 93E20.

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Figure 1. Trajectories of the type-1 agents' states when N = 500

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Figure 2. Trajectories of the type-2 agents' states when N = 500

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Figure 3. Trajectories of the type-1 agents state average and the mean field term

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Figure 4. Trajectories of the type-2 agents state average and the mean field term

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