Linear-Quadratic Mean-Field Game for Stochastic Systems with Partial Observation

TL;DR

This paper develops a decentralized control framework for large-population linear-quadratic stochastic systems with partial observations, providing explicit solutions and verifying equilibrium properties with an application in finance.

Contribution

It introduces a mean-field game approach with a backward separation principle for partial information, offering explicit control solutions and equilibrium analysis.

Findings

Explicit decentralized control solutions derived.

Verification of ε-Nash equilibrium property.

Application demonstrated in a financial example.

Abstract

This paper is concerned with a class of linear-quadratic stochastic large-population problems with partial information, where the individual agent only has access to a noisy observation process related to the state. The dynamics of each agent follows a linear stochastic differential equation driven by individual noise, and all agents are coupled together via the control average term. Using the mean-field game approach and the backward separation principle with a state decomposition technique, the decentralized optimal control can be obtained in the open-loop form through a forward-backward stochastic differential equation with the conditional expectation. The optimal filtering equation is also provided. By the decoupling method, the decentralized optimal control can also be further presented as the feedback of state filtering via the Riccati equation. The explicit solution of the control average limit is given, and the consistency condition system is discussed. Moreover, the related $\varepsilon$-Nash equilibrium property is verified. To illustrate the good performance of theoretical results, an example in finance is studied.