Probabilistic Theory of Mean Field Games, Volumes I & II
Probabilistic Theory of Mean Field Games, Volumes I & II
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平均场博弈的概率论，第一卷
DOI：
10.1090/noti2089
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发表时间：
2020
影响因子：
--
通讯作者：
J. Fouque
中科院分区：
文献类型：
--
作者：
J. Fouque
DOI：https://dx.doi.org/10.1090/noti2089控制，或数学金融，可以在流行的研究生教材中找到由Bracksendal [1]。在随机控制的背景下，一个单一的代理控制扩散过程的系数（漂移和最终的波动率）（如上所述），以优化目标函数。扩散过程的随机控制理论和应用以及与非线性二阶抛物型偏微分方程的关系已经得到了广泛的研究，现在已经成为概率论中的一个经典课题（参见Fleming和Soner的书[2]）。N人随机微分博弈的情况则完全不同。与单智能体随机控制的情况相反，N个智能体状态的动力学通常由伊藤型随机微分方程的耦合系统给出，该系统由布朗运动驱动，可能与普通噪声的情况相关。每个参与者，通过她的行动影响漂移和可能的波动条件，试图最小化/最大化她的目标函数的运行和终端成本/奖励。需要仔细定义行动/策略集的信息结构（开环、闭环等）来具体说明问题。有几种均衡的概念，它们中的每一个都缺乏唯一性，特别是当N很大时，很难计算。这些书都是关于纳什均衡的。纳什均衡是一组策略，如果任何一个参与者试图改变她的行动，其他人被固定，那么，她不能严格地更好。这种均衡也可以被理解为系统的最佳反应（参见Carmona [3]的书的第5章）。在2006年，Lasry和Lions（见[4]）提出了一种方法来获得大量玩家之间随机微分博弈的近似纳什均衡概率论平均场博弈与应用：第一卷，平均场FBSDES，控制和游戏;第二卷，平均场博弈与公共噪声和主方程
DOI: https://dx.doi.org/10.1090/noti2089 control, or mathematical finance, can be found in the popular graduate text by Øksendal [1]. In the context of stochastic control, one single agent controls the coefficients (the drift and eventually the volatility) of a diffusion process (as described above), in order to optimize an objective function. The theory and applications of stochastic control for diffusion processes and the relation with nonlinear second-order parabolic partial differential equations have been vastly studied and now form a classical topic in probability theory (see the book by Fleming and Soner [2]). The situation with N-player stochastic differential games is quite different. As opposed to the case of stochastic control with one agent, typically, the dynamics of the states of the N agents are given by a coupled system of stochastic differential equations of the Itô type driven by Brownian motions possibly correlated as in the case with a common noise. Each player, through her action affecting drift and possibly volatility terms, tries to minimize/maximize her objective function made of running and terminal costs/ rewards. The information structure of the set of actions/ strategies needs to be carefully defined (open-loop, closedloop, ...) in order to specify the problem. There are several notions of equilibria, each of them lacking uniqueness in general and difficult to compute in particular when N is large. The books under review are concerned with Nash equilibria. A Nash equilibrium is a set of strategies such that if any player tries to change her action, the others being fixed, then, she cannot be strictly better off. This equilibrium can also be understood as a best-response of the system (see Chapter 5 of the book by Carmona [3]). In 2006, Lasry and Lions (see [4]) proposed a methodology to obtain approximate Nash equilibria for stochastic differential games between a large number of players Probabilistic Theory of Mean Field Games with Applications: Volume I, Mean Field FBSDEs, Control, and Games; Volume II, Mean Field Games with Common Noise and Master Equations By René Carmona and François Delarue