Complex Stochastic Systems: Analysis, Control and Applications

本课题研究一类复杂随机系统及其相关的随机控制问题。底层系统受到各种随机力的影响。本项目的具体目标和预期结果如下：(1)研究具有跳变的状态切换扩散的稳定性和遍历性，并建立一个Feynman-Kac型公式；(2)发展状态切换跳跃扩散的奇异随机控制理论，并为相关控制问题设计可行有效的数值方案；(3)将理论成果应用于生物学、数学金融学和风险管理。这个项目的预期结果将有助于深入理解一类复杂的随机系统。反过来，这将促进此类系统在金融和生物学等领域的应用。状态切换扩散的正则和奇异混合控制问题可能产生许多新的和有趣的数学结果，以及随机分析、数值逼近和控制理论中的新问题。这个研究项目是由生态系统建模、金融工程、保险风险流程、制造和生产计划等新兴应用驱动的。这些系统的动力学不可避免地包含不确定性。例如，在生态系统建模中，一般生态系统的种群动态具有两个显著特征：(i)每天都有抖动，造成微小波动，以及流行病、地震和海啸等罕见事件造成的大量种群损失；（ii）由于许多物种的生长速度和承载能力经常根据营养、水供应和/或食物资源的变化而变化，该系统发生了质的变化。在保险风险过程的动态、风险资产的价格和其他方面也可以观察到类似的现象。这些特征使得文献中常用的模型不足以描述如此复杂的系统。所提议的项目旨在考虑这些固有的随机力量，并提出随机过程和相关的控制问题，这些问题在处理这些现实世界的应用时是通用的、灵活的，但在数学上是可处理的。它提出了一种新的用于复杂系统建模和分析的随机过程，获得了这种系统的长期行为，发展了奇异控制理论，并设计了控制问题的数值方案。学生培训和教育，学科和跨学科合作，以及通过出版物和演讲传播研究成果是这个项目的组成部分。

This research project considers a class of complex stochastic systems and related stochastic control problems. The underlying systems are subject to various random forces. The specific aims and anticipated results of this project are as follows: (1) to investigate stability and ergodicity and to establish a Feynman-Kac type formula for regime-switching diffusions with jumps; (2) to develop singular stochastic control theories for regime-switching jump diffusions and to design feasible and effective numerical schemes for the associated control problems; and (3) to apply the theoretical results to biology, mathematical finance, and risk management. The expected results of this project will contribute to an in-depth understanding of a wide class of complex stochastic systems. This, in turn, will facilitate the applications of such systems in areas such as finance and biology. The mixed regular and singular control problems for regime-switching diffusions are likely to generate many new and interesting mathematical results as well as new problems in stochastic analysis, numerical approximation and control theory.This research project is motivated by emerging applications arising from ecosystem modeling, financial engineering, insurance risk processes, manufacturing and production planning. The dynamics of these systems inevitably involve uncertainty. For example, in ecosystem modeling, the population dynamics of a general ecosystem possess two salient features: (i) there is day-to-day jitter that causes minor fluctuations as well as big population loss caused by rare events such as epidemics, earthquakes, and tsunamis; and (ii) there are qualitative changes in the system stemming from the fact that the growth rates and carrying capacities of many species often vary according to changes in nutrition, water supply, and/or food resources. Similar phenomena are observed in the dynamics of insurance risk processes, the price of a risky asset, and others. These features make the usual models in the literature inadequate in describing such complex systems. The proposed project aims to take into these inherent random forces and propose stochastic processes and related control problems that are general and flexible, yet mathematically tractable, in dealing with these real-world applications. It presents novel stochastic processes for modeling and analysis of complex systems, obtains long-time behavior of such systems, develops singular control theories, and designs numerical schemes for the control problems. Student training and education, disciplinary and interdisciplinary collaborations, and the dissemination of research results through publications and presentations are integral parts of this project.