Linear-Quadratic Large-Population Problem with Partial Information: Hamiltonian Approach and Riccati Approach

TL;DR

This paper develops explicit decentralized strategies for large-population linear-quadratic mean-field games under partial information, using Hamiltonian and Riccati approaches, and verifies their equilibrium properties.

Contribution

It introduces a novel combination of Hamiltonian and Riccati methods to solve partial information mean-field games with control constraints and provides explicit strategies and equilibrium analysis.

Findings

Explicit decentralized strategies derived for constrained and unconstrained cases.

Proved well-posedness of nonlinear mean-field FBSDEs with projection.

Validated strategies through an inter-bank borrowing and lending application.

Abstract

This paper studies a class of partial information linear-quadratic mean-field game problems. A general stochastic large-population system is considered, where the diffusion term of the dynamic of each agent can depend on the state and control. We study both the control constrained case and unconstrained case. In control constrained case, by using Hamiltonian approach and convex analysis, the explicit decentralized strategies can be obtained through projection operator. The corresponding Hamiltonian type consistency condition system is derived, which turns out to be a nonlinear mean-field forward-backward stochastic differential equation with projection operator. The well-posedness of such kind of equations is proved by using discounting method. Moreover, the corresponding $\varepsilon$-Nash equilibrium property is verified. In control unconstrained case, the decentralized strategies can be further represented explicitly as the feedback of filtered state through Riccati approach. The existence and uniqueness of a solution to a new Riccati type consistency condition system is also discussed. As an application, a general inter-bank borrowing and lending problem is studied to illustrate that the effect of partial information cannot be ignored.