Stochastic Analysis and Numerics for Large Scale Dynamical Systems, with Applications

银行系统和其他受随机效应、不完整或不完全观察和系统性风险影响的大型网络日益复杂，对新的、有效的数学工具的分析需求日益增加。该奖项针对“大规模”动力系统的随机和计算分析方面的三个项目，因为它们被设置在非常高维的空间中，或者涉及系统中大量参与主体的相互作用。第一个项目将侧重于开发具有大量空间维度的偏微分方程（PDEs）的有效数值格式。这些方案可能对面向计算的金融工具产生重大影响，例如涉及非常大的投资组合的基于模型的交易算法。第二个项目和第三个项目分别研究以集中（平均场）和非集中（网络）方式相互作用的大型金融系统的结构。研究结果将为个人投资者和监管机构提供衡量和管理系统性风险的工具，尤其是大规模银行间贷款市场的违约传染。该奖项还将导致研究生的参与和培训，并通过期刊出版物和会议进行传播。第一个项目将为高维偏微分方程和路径相关偏微分方程开发有效的蒙特卡罗方法。当方程的维度被允许在数百或更高时，这些方案预计将是有效的，因此基本上打破了臭名昭著的“维度诅咒”，这个诅咒已经困扰了数值分析师和计算机科学家几十年。第二个项目在主方程及其变化的总体框架下探讨随机博弈/控制理论和定量金融中的各种理论问题和几个实际问题。该项目将利用最近开发的几种技术工具，包括时间一致性原则和动态效用方法。第三个关于动态系统风险的项目研究了一个复杂的网络，其中银行之间的传染效应以时间一致和非集中的方式描述。通过考虑概率测度的状态空间，网络的动态运动可以用测度值sde来描述，而系统风险可以用一定的主方程来表征。拟议研究中的所有项目都与应用领域有直接或间接的联系，特别是与随机金融/精算科学。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 increasing complexity of banking systems and other large-scale networks subject to random effects, incomplete or partial observations, and systemic risk, put increasing demands for new and efficient mathematical tools for their analysis. This award addresses three projects in the stochastic and computational analysis of "large-scale" dynamical systems, in the sense that they are set in spaces of very high dimensions or involve the interactions of a large number of participating agents in the system. The first project will focus on the development of efficient numerical schemes for partial differential equations (PDEs) with a large number of spatial dimensions. These schemes could have a significant impact on computation-oriented financial instruments, such as model-based trading algorithms involving very-large portfolios. The second project and third projects study the structures of large financial systems that are interacting in a centralized (mean-field) and non-centralized (network) manner, respectively. The results will provide tools for the measurement and management of systemic risk, particularly default contagion for large scale interbank lending markets, both for individual investors and for regulators. The award will also result in the involvement and training of graduate students and dissemination through journal publications and conferences. The first project will develop efficient Monte-Carlo methods for high-dimensional PDEs and path-dependent PDEs (PPDEs). These schemes are expected to be efficient when the dimension of the equations is allowed to be in the hundreds or higher, hence essentially breaking the notorious "curse of dimensionality" that has been baffling the numerical analysts and computer scientists for decades. The second project explore various theoretical issues as well as several practical problems in stochastic game/control theory and quantitative finance under the general framework of master equations and their variations. This project will utilize several recently developed technical tools, including the time consistency principle and the dynamic utility approach. The third project on dynamic systemic risk investigates a complex network in which the contagion effect among banks are described in a time-consistent, and non-centralized manner. By considering the state space of probability measures, the dynamical movements of networks can be described by measure-valued SDEs, while the systemic risk is characterized by certain master equations. All projects in the proposed research have direct or indirect connections to applied fields, especially to stochastic finance/actuarial sciences.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.