标题： 在共同噪声不确定性下的鲁棒均场控制 显示英文标题

标题： Robust mean-field control under common noise uncertainty

摘要： 我们提出并分析了一个框架，用于在共同噪声不确定性下的离散时间鲁棒均场控制问题。 在这个框架中，均场相互作用描述了无限多个合作代理的状态和动作的集体行为，而共同噪声——一种影响所有代理状态动态的随机扰动——是不确定的。 一个社会规划者在无限时间范围内对开环控制进行优化，以最大化代表性代理的最坏情况期望回报，其中最坏情况对应于所有导致共同噪声过程未知真实定律的候选概率测度中最不利的概率测度。 我们将这种优化称为在共同噪声不确定性下的鲁棒均场控制问题。 我们首先表明，这个问题作为合作$N$-代理鲁棒优化问题的渐近极限出现，通常被称为混沌传播。 然后，通过将鲁棒均场控制问题与概率测度空间上的提升鲁棒马尔可夫决策问题相关联，并为提升的鲁棒马尔可夫决策问题建立动态规划原理和贝尔曼-伊萨克斯固定点定理，证明了最优开环控制的存在性。 最后，我们通过受金融中的分布规划和系统性风险启发的数值实验来补充我们的理论结果，突出了考虑共同噪声不确定性的优势。 显示英文摘要

摘要： 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.