New Problems in Mean Field Control Theory

平均场游戏于2006年由Lasry-Lions和Huang-Caines-Malhamé独立推出。他们的想法是将平均场方法从物理学和力学应用于社会科学。平均场方法允许描述流体和气体的宏观行为，而不考虑单个粒子的运动。我们可以考虑一个代表性的粒子，它的动力学受到平均场项的影响，平均场项代表了来自所有其他粒子的平均效应。在社会科学中，粒子被代理人取代。例如，在对高速公路上的交通进行建模时，智能体可以是汽车。与物理学的一个重要区别是，代理人是决策者，动态对应于决策。该建议为这些平均场模型开发了控制方法，称为平均场控制理论。结果将应用于动态网络，并为经济理论等应用提供见解。平均场控制理论的一个基本问题是求解Hamilton-Jacobi-Bellman和Fokker Planck（HJB-FP）方程组。由Lions引入的主方程允许通过驱动解耦参数将HJB-FP方程系统封装在单个方程中。一个严重的困难是，一个人需要与概率措施的功能。狮子介绍的想法，一个可以写的主方程使用的空间参数的希尔伯特空间的平方可积随机变量。这导致了新的控制问题，在这个特殊的希尔伯特空间，这是研究的第一个目标。第二个目标是研究平均场控制理论中的具体问题，其动机是经济学。

Mean field games were introduced in 2006, independently by Lasry-Lions and Huang-Caines-Malhamé. Their idea was to adapt the mean field approach from physics and mechanics to social sciences. The mean field approach allows the description of the macroscopic behavior of fluids and gases without considering the motion of the individual particles. One can consider a representative particle whose dynamics is influenced by a mean field term that represents an averaging effect coming from all other particles. In the social sciences, particles are replaced with agents. For example, an agent can be a car when modeling traffic on highways. An important difference from physics is that the agents are decision makers and the dynamics correspond to decisions. This proposal develops control methods for these mean field models, called Mean Field Control Theory. Results will be applied to dynamic networks and provide insight for applications such as economic theory.A basic problem in mean field control theory is to solve a system of Hamilton-Jacobi-Bellman and Fokker Planck (HJB-FP) equations. The Master Equation, introduced by Lions, allows one to encapsulate the system of HJB-FP equations in a single equation by driving a decoupling argument. A serious difficulty is the fact that one needs to work with functions of probability measures. Lions introduced the idea that one could write the Master Equation using for the space of arguments the Hilbert space of square integrable random variables. This leads to new control problems in this particular Hilbert space, which is the first objective of the research. The second objective is to study specific problems in mean field control theory, motivated by economics.