Dynamical Approaches for Some Complex Stochastic Systems

一些复杂随机系统的动力学方法

• 批准号: 2205972
• 负责人: Jianfeng Zhang
• 金额: $ 32万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205972&HistoricalAwards=false
• 关键词: Dynamical; Approaches; Some; Complex; Stochastic

The principal investigators will undertake several research projects with a special focus on dynamic optimization/game problems involving complex stochastic systems, as well as effective mathematical tools for solving these problems. The research on the general mean field master equations introduces new methodologies that will lead to a powerful tool for studying large interacting particle systems appearing in numerous applications in economics, finance (especially systemic risk), social science, and engineering. The research concerning the set-valued stochastic dynamical systems develops some new technical tools and has intrinsic connection to front propagation and mean curvature motions. Some new concepts in the project are expected to fundamentally change the current framework of set-valued stochastic analysis. The part concerning the Kyle-Back strategic insider equilibrium model brings new perspectives to an important problem in the market macrostructure theory as well as the concept of dynamic Markov bridges. All the projects that will be pursued have direct connections to applied fields such as stochastic control/game and stochastic finance/economics. The PIs will continue actively involving Ph.D students and postdoc fellows in research, disseminating research findings through conferences, and strengthening the connections with local financial communities through a colloquium series.The first part of the research introduces new methodologies to find the crucial monotonicity conditions for general mean field games, which will ensure the uniqueness of the mean field equilibrium and lead to the global well-posedness of the associated master equation, an infinite dimensional PDE or PDE system that characterizes the dynamics of the value function. The second part of the research focuses on the characterization of stochastic dynamic systems whose values are “sets”, motivated by several signature cases including the problems in the first part when uniqueness of equilibria fails, the time-inconsistent problems studied in previous research, and the issue of dynamic multivariate (systemic) risk measures. The new theory of set-valued PDEs and set-valued Backward SDEs, along with several new notions such as Itô’s formula for set-valued functions and a new set-valued stochastic integral desirable for the set-valued martingale representation will be considered, and are expected to have fundamental impact to the existing set-valued stochastic analysis. The third part of the research concerns the Kyle-Back equilibrium model with dynamic information. A new stochastic two-point boundary value problem will be considered, as a theoretical basis for finding the equilibrium under a fairly general model of underlying assets that allows interaction among the different agents in the market.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.

主要研究人员将开展几个研究项目，特别关注涉及复杂随机系统的动态优化/博弈问题，以及解决这些问题的有效数学工具。一般平均场主方程的研究引入了新的方法，这将导致一个强大的工具，用于研究大型相互作用粒子系统出现在经济学，金融（特别是系统风险），社会科学和工程的众多应用。对集值随机动力系统的研究发展了一些新的技术工具，并与波前传播和平均曲率运动有着内在的联系。该项目中的一些新概念有望从根本上改变目前的集值随机分析框架。Kyle-Back战略内部人均衡模型的提出，为市场宏观结构理论中的一个重要问题以及动态马尔可夫桥的概念提供了新的视角。 所有将被追求的项目都与随机控制/游戏和随机金融/经济学等应用领域有直接联系。研究所将继续积极邀请博士生和博士后研究员参与研究，通过会议传播研究成果，并通过一系列研讨会加强与当地金融界的联系。研究的第一部分介绍了寻找一般平均场博弈的关键单调性条件的新方法，这将确保平均场平衡的唯一性，并导致相关主方程的全局适定性，无限维PDE或PDE系统，其特征在于值函数的动力学。第二部分的研究重点是随机动态系统的特征，其值是“集”，由几个签名的情况下，包括在第一部分的问题时，唯一的平衡失败，在以前的研究中研究的时间不一致的问题，和动态多变量（系统）风险措施的问题。新的集值偏微分方程和集值倒向偏微分方程的理论，沿着几个新的概念，如伊藤公式的集值函数和一个新的集值随机积分集值鞅表示所需的将被考虑，并预计将有根本性的影响，现有的集值随机分析。第三部分研究动态信息下的Kyle-Back均衡模型。一个新的随机两点边值问题将被考虑，作为一个理论基础，在一个相当普遍的基础资产模型下寻找均衡，该模型允许市场上不同代理人之间的相互作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Mean field games of controls: Propagation of monotonicities

• DOI: 10.3934/puqr.2022015
• 发表时间: 2022-05
• 期刊: Probability, Uncertainty and Quantitative Risk
• 作者: Chenchen Mou;Jianfeng Zhang

A general conditional McKean–Vlasov stochastic differential equation

一般条件 McKean Vlasov 随机微分方程

• DOI: 10.1214/22-aap1858
• 发表时间: 2023
• 期刊: The Annals of Applied Probability
• 作者: Buckdahn, Rainer;Li, Juan;Ma, Jin
• 通讯作者: Ma, Jin