英文标题：
《The Master Equation in Mean Field Theory》
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作者：
Alain Bensoussan, Jens Frehse, Phillip Yam
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最新提交年份：
2014
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英文摘要：
In his lectures at College de France, P.L. Lions introduced the concept of Master equation, see [5] for Mean Field Games. It is introduced in a heuristic fashion, from the system of partial differential equations, associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in[2], consists in a formal analogy of terms. The interest of this equation is that it contains interesting particular cases, which can be studied directly, in particular the system of HJB-FP (Hamilton-Jacobi-Bellman, Fokker-Planck) equations obtained as the limit of the finite Nash equilibrium game, when the trajectories are independent, see [4]. Usually, in mean field theory, one can bypass the large Nash equilibrium, by introducing the concept of representative agent, whose action is influenced by a distribution of similar agents, and obtains directly the system of HJB-FP equations of interest, see for instance [1]. Apparently, there is no such approach for the Master equation. We show here that it is possible. We first do it for the Mean Field type control problem, for which we interpret completely the Master equation. For the Mean Field Games itself, we solve a related problem, and obtain again the Master equation.
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中文摘要：
在法兰西学院的演讲中，P.L.Lions介绍了主方程的概念，参见[5]中的平均场游戏。它是以一种启发式的方式从偏微分方程组中引入的，与大量但有限的参与者的纳什均衡有关。该方法（也在[2]中解释）包括术语的形式类比。该方程的有趣之处在于，它包含了可以直接研究的有趣的特殊情况，尤其是HJB-FP（Hamilton Jacobi Bellman，Fokker-Planck）方程系统，该系统作为有限纳什均衡博弈的极限，当轨迹独立时，参见【4】。通常，在平均场理论中，可以通过引入代表性代理的概念，绕过大型纳什均衡，代表性代理的行为受类似代理分布的影响，并直接获得感兴趣的HJB-FP方程组，参见示例【1】。显然，对于主方程没有这样的方法。我们在这里表明这是可能的。我们首先对平均场型控制问题这样做，我们完全解释了主方程。对于平均场对策本身，我们解决了一个相关问题，并再次得到主方程。
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分类信息：

一级分类：Mathematics 数学
二级分类：Analysis of PDEs 偏微分方程分析
分类描述：Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE\'s, conservation laws, qualitative dynamics
存在唯一性，边界条件，线性和非线性算子，稳定性，孤子理论，可积偏微分方程，守恒律，定性动力学
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一级分类：Mathematics 数学
二级分类：Dynamical Systems 动力系统
分类描述：Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations
微分方程和流动的动力学，力学，经典的少体问题，迭代，复杂动力学，延迟微分方程
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一级分类：Mathematics 数学
二级分类：Optimization and Control 优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance 数量金融学
二级分类：Mathematical Finance 数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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