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.

首席研究人员将特别关注涉及复杂随机系统的动态优化/游戏问题，以及解决这些问题的有效数学工具，以特别关注动态优化/游戏问题。对一般平均野外方程式的研究介绍了新方法，该方法将导致一种强大的工具，用于研究在经济学，金融（尤其是系统性风险），社会科学和工程中出现在许多应用中的大型相互作用粒子系统。关于设定的随机动态系统的研究开发了一些新的技术工具，并与前繁殖和平均曲率运动具有内在的联系。预计该项目中的一些新概念将从根本上改变设定值随机分析的当前框架。关于凯尔后卫战略内幕均衡模型的部分为市场宏观结构理论以及动态马尔可夫桥的概念带来了新的观点。所有将要进行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的联系。 PI将继续积极地涉及博士学位学生和博士学位研究员参与研究，通过会议传播研究结果，并通过座谈会系列加强与当地金融社区的联系。研究的第一部分介绍了新方法，介绍了将一般平均值的至关重要的单调性介绍的新方法，这将确保一般平均水平的独特性，并确保全球平均水平，并置于全球范围内，并引入全球平均水平，并导致全球范围的平均范围，并导致全球范围内的平均水平。表征值函数的动力学的尺寸PDE或PDE系统。研究的第二部分重点是对随机动态系统的表征，其价值是“集合”的表征，这些系统是由几个签名案例进行的，包括第一部分中的问题，当平衡的唯一性失败，以前的研究中研究的时间不一致的问题以及动态多变量（全身）风险测量问题。设置值PDE和设置值的向后SDE的新理论以及几个新的注释，例如ITô的设置值功能的公式以及对设置价值的Martingale代表所需的新的设置值的随机积分，并预计将对现有的设置值分析产生根本性的影响。研究的第三部分涉及使用动态信息的Kyle-Back平衡模型。将考虑一个新的随机两点边界价值问题，是在相当普遍的基础资产模型下找到平衡的理论基础，该模型允许市场中不同的代理商之间的互动。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子和更广泛的影响来评估的支持，并被视为珍贵的支持。

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