Mean Field Games and Master equations
平均场游戏和主方程
基本信息
- 批准号:EP/X020320/1
- 负责人:
- 金额:$ 38.82万
- 依托单位:
- 依托单位国家:英国
- 项目类别:Research Grant
- 财政年份:2023
- 资助国家:英国
- 起止时间:2023 至 无数据
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
There are different approaches to model a large collection of interacting particles or agents. One such approach is based on a microscopical viewpoint, where one wants to determine the attributes (such as position, velocity, etc.) of each individual particle at any given time. A possible way to mathematically study such an approach would be by a system of coupled ordinary/stochastic differential equations, having these attributes as unknowns. However, implementing numerical solvers that are based on this approach is in general quite costly, and many times is even impossible, given the huge number (billions and billions) of unknowns. A macroscopic approach, that mainly arose in the framework of statistical physics, uses a so-called 'mean field perspective'. This in general aims to describe the behaviour of the particles via the time evolution of their density (i.e. as a 'cloud'). Such models typically lead to partial differential equations for such macroscopic quantities as unknowns. Since their introduction (initiated around 2006, independently by Caines--Huang--Malhamé, and Lasry--Lions), the theory of mean field games have found multiple applications both in pure and applied mathematics. It studies strategic decision making in large populations where the individual agents interact via certain mean-field quantities (through the density, velocities, controls, etc. of the other agents). It provides powerful tools for applications ranging from quantum mechanics to biodiversity ecology and it has already had a significant impact on models in social sciences, macroeconomics, stock markets, risk management and wealth distribution, and on biological systems. This theory has its roots in the mean field theory from statistical physics, however, in mean field games the agents would like to find optimal strategies. This is somehow in contrast with physical models, where particles are typically governed by the laws of nature. So, in mean field games, the general goal is to find and characterize Nash-type equilibrium configurations. The master equation in mean field games was introduced by P.-L. Lions and it represents the heart of the theory. This is an infinite dimensional nonlocal Hamilton--Jacobi--Bellman equation on the space of probability measures and it encodes the Nash equilibria in mean field games. Among others, it serves as a powerful tool to prove and quantify the mean field limit and propagation of chaos for stochastic games when the number of agents tends to infinity, which is important in applications as well as of great interest theoretically. The question of solvability of the master equation initiated an important programme and outstanding open problems in the field. By the nature of this equation, it is expected that in general classical solutions will break down in finite time. So, its solvability was established either for short time horizon or under special structural conditions on the data that satisfy the so-called Lasry--Lions monotonicity condition. Also, most of the results in the literature use the regularisation effect of a non-degenerate idiosyncratic noise.In this proposal, we will study degenerate master equations in the lack of such a regularisation effect or structural assumptions imposed by the Lasry--Lions monotonicity condition. Instead, we will rely on the so-called displacement monotonicity condition, that stems from the notion of displacement convexity arising in the theory of optimal transport. This condition allows us to investigate a general class of degenerate models (that are sometimes closer to real life applications), by a unified way. Roughly speaking, displacement monotonicity helps also to restore important regularity properties, which were lost in the lack of the non-degenerate noise. This proposal includes both purely deterministic models and models subject to common noise.
有不同的方法来模拟大量的相互作用的粒子或代理。一种这样的方法是基于微观视角,其中想要确定属性(诸如位置、速度等)。每一个粒子的运动轨迹数学研究这种方法的一种可能方法是通过耦合的普通/随机微分方程系统,将这些属性作为未知数。然而,实现基于这种方法的数值求解器通常是相当昂贵的,并且考虑到大量(数十亿)的未知数,很多时候甚至是不可能的。一种宏观方法,主要出现在统计物理的框架,使用所谓的“平均场视角”。这通常旨在通过粒子密度的时间演变来描述粒子的行为(即作为“云”)。这样的模型通常导致这样的宏观量作为未知量的偏微分方程。自2006年左右由Caines-Huang-Malhamé和Lasry-Lions独立提出以来,平均场博弈理论在纯数学和应用数学中都有广泛的应用。它研究大群体中的战略决策,其中个体代理通过某些平均场量(通过其他代理的密度,速度,控制等)进行交互。它为从量子力学到生物多样性生态学的应用提供了强大的工具,并且已经对社会科学,宏观经济学,股票市场,风险管理和财富分配以及生物系统的模型产生了重大影响。该理论起源于统计物理学的平均场理论,然而,在平均场游戏中,代理人希望找到最优策略。这在某种程度上与物理模型相反,在物理模型中,粒子通常受自然定律的支配。因此,在平均场博弈中,总的目标是找到并描述纳什型均衡配置。平均场博弈中的主方程是由P. L.它代表了理论的核心。这是概率测度空间上的一个无限维非局部汉密尔顿-雅可比-贝尔曼方程,它编码了平均场博弈中的纳什均衡。其中,它作为一个强有力的工具,证明和量化的平均场极限和混沌的传播的随机博弈时,代理人的数量趋于无穷大,这是重要的应用程序,以及极大的理论兴趣。问题的可解性的主方程发起了一个重要的方案和悬而未决的开放问题领域。根据这个方程的性质,一般来说,经典解会在有限时间内崩溃。因此,它的可解性是建立在短时间范围内或在特殊的结构条件下的数据,满足所谓的Lasry-Lions单调性条件。此外,大多数文献中的结果使用的正则化效应的非退化的特质noise.In这个建议中,我们将研究退化的主方程在缺乏这样的正则化效应或结构假设所施加的Lasry-Lions单调性条件。相反,我们将依赖于所谓的位移单调性条件,它源于最优运输理论中的位移凸性概念。这个条件允许我们通过统一的方式研究一般类的退化模型(有时更接近于真实的生活应用)。粗略地说,位移单调性也有助于恢复重要的规律性属性,这些属性在缺乏非退化噪声的情况下丢失。该建议包括纯确定性模型和受共同噪声影响的模型。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
数据更新时间:{{ journalArticles.updateTime }}
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
数据更新时间:{{ journalArticles.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ monograph.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ sciAawards.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ conferencePapers.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ patent.updateTime }}
Alpar Meszaros其他文献
Alpar Meszaros的其他文献
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
相似国自然基金
Graphon mean field games with partial observation and application to failure detection in distributed systems
- 批准号:
- 批准年份:2025
- 资助金额:0.0 万元
- 项目类别:省市级项目
Research on Quantum Field Theory without a Lagrangian Description
- 批准号:24ZR1403900
- 批准年份:2024
- 资助金额:0.0 万元
- 项目类别:省市级项目
Development of a Linear Stochastic Model for Wind Field Reconstruction from Limited Measurement Data
- 批准号:
- 批准年份:2020
- 资助金额:40 万元
- 项目类别:
新型Field-SEA多尺度溶剂模型的开发与应用研究
- 批准号:21506066
- 批准年份:2015
- 资助金额:21.0 万元
- 项目类别:青年科学基金项目
相似海外基金
Risk-Sensitivity in Mean Field Games and Energy Markets
平均场博弈和能源市场的风险敏感性
- 批准号:
RGPIN-2022-05337 - 财政年份:2022
- 资助金额:
$ 38.82万 - 项目类别:
Discovery Grants Program - Individual
Asymptotic analysis on PDEs appearing in mean field games, crystal growth and anomalous diffusion
平均场博弈、晶体生长和反常扩散中偏微分方程的渐近分析
- 批准号:
22K03382 - 财政年份:2022
- 资助金额:
$ 38.82万 - 项目类别:
Grant-in-Aid for Scientific Research (C)
Systems, Control and Mean Field Games on Networks
网络上的系统、控制和平均场博弈
- 批准号:
RGPIN-2019-05336 - 财政年份:2022
- 资助金额:
$ 38.82万 - 项目类别:
Discovery Grants Program - Individual
Interacting Particle Systems and Mean-field games Workshops
交互粒子系统和平均场游戏研讨会
- 批准号:
2207572 - 财政年份:2022
- 资助金额:
$ 38.82万 - 项目类别:
Standard Grant
Risk-Sensitivity in Mean Field Games and Energy Markets
平均场博弈和能源市场的风险敏感性
- 批准号:
DGECR-2022-00468 - 财政年份:2022
- 资助金额:
$ 38.82万 - 项目类别:
Discovery Launch Supplement
Mean Field Games, Information Design and Evolutionary Finance
平均场博弈、信息设计和进化金融
- 批准号:
RGPIN-2020-06290 - 财政年份:2022
- 资助金额:
$ 38.82万 - 项目类别:
Discovery Grants Program - Individual
Mean Field Games, Information Design and Evolutionary Finance
平均场博弈、信息设计和进化金融
- 批准号:
RGPIN-2020-06290 - 财政年份:2021
- 资助金额:
$ 38.82万 - 项目类别:
Discovery Grants Program - Individual
CAREER: Mean Field Games with Economics Applications: New Techniques in Partial Differential Equations
职业:平均场博弈与经济学应用:偏微分方程新技术
- 批准号:
2045027 - 财政年份:2021
- 资助金额:
$ 38.82万 - 项目类别:
Continuing Grant
CAREER: Stochastic Games on Large Graphs in the Mean Field Regime and Beyond
职业:平均场制度及其他大图上的随机博弈
- 批准号:
2045328 - 财政年份:2021
- 资助金额:
$ 38.82万 - 项目类别:
Continuing Grant
Systems, Control and Mean Field Games on Networks
网络上的系统、控制和平均场博弈
- 批准号:
RGPIN-2019-05336 - 财政年份:2021
- 资助金额:
$ 38.82万 - 项目类别:
Discovery Grants Program - Individual