Infinite-Dimensional LQ Mean Field Games with Common Noise: Small and Arbitrary Finite Time Horizons

2026-01-20

AI Summary

This study addresses the existence and uniqueness of equilibrium solutions for linear-quadratic mean-field games in Hilbert spaces under common noise, with a particular focus on well-posedness over arbitrary finite time horizons. By modeling the common noise via an infinite-dimensional Wiener process, the authors formulate the mean-field consistency condition as a coupled forward–backward stochastic evolution system and analyze it within the mild solution framework using stochastic evolution equation theory. The main contribution lies in establishing, for the first time, a well-posedness theory for such coupled systems over any finite time interval, thereby overcoming previous restrictions to small time horizons. The work not only proves existence, uniqueness, and the ε-Nash equilibrium property of the solution but also extends these results to general finite time domains, ensuring global well-posedness of the system.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems

Abstract

We extend the results of (Liu and Firoozi, 2025), which develops the theory of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces, by incorporating a common noise. This common noise is modeled as an infinite-dimensional Wiener process affecting the dynamics of all agents. In the presence of common noise, the mean-field consistency condition is characterized by a system of coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces, whereas, in its absence it is represented by coupled forward-backward deterministic evolution equations. We establish the existence and uniqueness of solutions to the coupled linear FBSEEs associated with the LQ MFG framework for small time horizons and prove the $\epsilon$-Nash property of the resulting equilibrium strategy. Furthermore, we establish the well-posedness of these coupled linear FBSEEs for arbitrary finite time horizons. Beyond the specific context of MFGs, our analysis also yields a broader contribution by providing, to the best of our knowledge, the first well-posedness result for a class of infinite-dimensional linear FBSEEs, for which only mild solutions exist, over arbitrary finite time horizons.

Problem

• Mean Field Games
• Common Noise
• Hilbert Spaces
• Forward-Backward Stochastic Evolution Equations
• Linear-Quadratic

Innovation

• Mean Field Games
• Common Noise
• Hilbert Spaces
• Forward-Backward Stochastic Evolution Equations
• Well-posedness