Robust mean-field control under common noise uncertainty

TL;DR

This paper develops a framework for robust mean-field control under uncertain common noise, providing theoretical foundations and numerical insights for systems with collective behavior and shared disturbances.

Contribution

It introduces a novel robust mean-field control model under common noise uncertainty, establishing existence of optimal controls and linking to a lifted robust MDP framework.

Findings

Optimal controls exist under the proposed framework.

The approach effectively handles worst-case common noise scenarios.

Numerical experiments demonstrate practical advantages in finance and distribution planning.

Abstract

We propose and analyze a framework for discrete-time robust mean-field control problems under common noise uncertainty. In this framework, the mean-field interaction describes the collective behavior of infinitely many cooperative agents' state and action, while the common noise -- a random disturbance affecting all agents' state dynamics -- is uncertain. A social planner optimizes over open-loop controls on an infinite horizon to maximize the representative agent's worst-case expected reward, where worst-case corresponds to the most adverse probability measure among all candidates inducing the unknown true law of the common noise process. We refer to this optimization as a robust mean-field control problem under common noise uncertainty. We first show that this problem arises as the asymptotic limit of a cooperative $N$-agent robust optimization problem, commonly known as propagation of chaos. We then prove the existence of an optimal open-loop control by linking the robust mean field control problem to a lifted robust Markov decision problem on the space of probability measures and by establishing the dynamic programming principle and Bellman--Isaac fixed point theorem for the lifted robust Markov decision problem. Finally, we complement our theoretical results with numerical experiments motivated by distribution planning and systemic risk in finance, highlighting the advantages of accounting for common noise uncertainty.