Indefinite Linear-Quadratic Partially Observed Mean-Field Game

TL;DR

This paper develops a comprehensive framework for indefinite linear-quadratic mean-field games with partial observations and common noise, deriving decentralized strategies and proving their equilibrium properties.

Contribution

It introduces a novel model incorporating indefinite cost matrices, general stochastic observations, and establishes the well-posedness and equilibrium of decentralized strategies.

Findings

Derived optimal decentralized strategies using Hamiltonian approach.

Proved the strategies form an ε-Nash equilibrium.

Applied the framework to a mean-variance portfolio problem.

Abstract

This paper investigates an indefinite linear-quadratic partially observed mean-field game with common noise, incorporating both state-average and control-average effects. In our model, each agent's state is observed through both individual and public observations, which are modeled as general stochastic processes rather than Brownian motions. {It is noteworthy that} the weighting matrices in the cost functional are allowed to be indefinite. We derive the optimal decentralized strategies using the Hamiltonian approach and establish the well-posedness of the resulting Hamiltonian system by employing a relaxed compensator. The associated consistency condition and the feedback representation of decentralized strategies are also established. Furthermore, we demonstrate that the set of decentralized strategies form an $\varepsilon$-Nash equilibrium. As an application, we solve a mean-variance portfolio selection problem.