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

点击翻译按钮获取中文摘要

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.