Dynamical Approaches for Some Complex Stochastic Systems

主要研究人员将承担几个研究项目，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。对一般平均场主方程的研究引入了新的方法，这些方法将成为研究大型相互作用粒子系统的有力工具，这些系统在经济、金融（特别是系统风险）、社会科学和工程领域的许多应用中都出现了。集值随机动力系统的研究发展了一些新的技术工具，并与前传播和平均曲率运动有着内在的联系。项目中的一些新概念有望从根本上改变集值随机分析的现有框架。Kyle-Back战略内部均衡模型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。所有将进行的项目都与随机控制/博弈和随机金融/经济学等应用领域有直接联系。各专业学院将继续积极邀请博士生及博士后参与研究，透过会议传播研究成果，并举办一系列学术研讨会，加强与本地金融界的联系。研究的第一部分介绍了寻找一般平均场对策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，即具有值函数动态特征的无限维PDE或PDE系统。第二部分研究的重点是值为“集合”的随机动态系统的特征，包括第一部分中均衡唯一性失效的问题、之前研究的时间不一致问题以及动态多变量（系统）风险度量问题。本文将考虑集值偏微分方程和集值后向偏微分方程的新理论，以及一些新概念，如Itô的集值函数公式和集值鞅表示所需的集值随机积分，并期望对现有的集值随机分析产生根本性的影响。研究的第三部分是动态信息的Kyle-Back均衡模型。本文将考虑一个新的随机两点边值问题，作为在允许市场中不同主体之间相互作用的相当一般的标的资产模型下寻找均衡的理论基础。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.