New Extensions of the Master Equation in Mean Field Control Theory and Applications

平均场对策和平均场类型控制理论在不确定条件下的决策和动态系统的控制方面取得了重大进展。当P.-L.Lions引入了一个被称为主方程的新的偏微分方程式时，迈出了重要的一步，它在概念上综合了该领域所有可能的方法。不幸的是，这是一个异常困难的方程式。在这个项目中，研究人员将开发新的数学技术来研究主方程。因此，这项工作是对平均场理论的概念核心-主方程的贡献，专注于两个重要的应用：能源市场和部分信息随机控制。这项工作更广泛的意义在于，它将对经济、金融、机器学习等领域产生新的见解。研究生和本科生将接受各种问题和概念的培训，并有机会进行研究和应用。项目的第一个问题需要扩展平均场控制，其中扩展不仅在于状态的概率分布，而且还在于控制的概率分布进入模型，和/或在代价泛函中最小化。它是一般理论的一个新分支，研究人员将把它发展并应用于能源市场和相关问题。第二个问题涉及非局部相关性，研究人员将把它应用于具有部分信息的随机控制领域，从平均场的角度提供新的见解。涉及的数学技术将包括无限维空间上的哈密尔顿-雅可比方程、倒向随机偏微分方程组和非局部方程的向前-向后系统。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Mean Field Games and Mean Field Type Control Theory have brought major advances in decision making under uncertainty and control of dynamic systems. A significant step was taken when P.-L. Lions introduced a new partial differential equation known as the Master Equation, which conceptually synthesizes all potential approaches to the field. Unfortunately, this is an extraordinarily difficult equation. In this project, the investigators will develop new mathematical techniques to study the Master Equation. Therefore, the work is a contribution to the conceptual core of mean field theory, the master equation, focusing on two important applications: energy markets and stochastic control with partial information. The broader significance of this work is that it will produce new insights into economics, finance, machine learning, and other fields. Students at both graduate and undergraduate levels will be trained in a large variety of issues and concepts, with opportunities for research and applications. The first problem of the project requires extended mean field control, where the extension lies in the fact that not only the probability distribution of the state, but also the probability distribution of the control enters in the model, and/or in the cost functional to be minimized. It is a new branch of the general theory, which the investigators will develop and apply to energy markets and related problems. The second problem regards nonlocal dependence, which the investigators will apply to the well-established domain of stochastic control with partial information, providing new insights from a mean field point of view. The mathematical techniques involved will include Hamilton-Jacobi equations on infinite dimensional spaces, backward stochastic partial differential equations, and forward-backward systems of nonlocal equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.