Complex Stochastic Systems: Analysis, Control and Applications

复杂随机系统：分析、控制和应用

• 批准号: 1108782
• 负责人: Chao Zhu
• 金额: $ 11.5万
• 依托单位: University of Wisconsin-Milwaukee
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108782&HistoricalAwards=false
• 关键词: Complex; Stochastic; Systems; Analysis; Control

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.

本研究计画考虑一类复杂随机系统及其相关的随机控制问题。基础系统受到各种随机力的影响。本课题的具体目标和预期成果如下：（1）研究带跳跃的状态转换扩散的稳定性和遍历性，建立Feynman-Kac型公式：（2）发展状态转换跳跃扩散的奇异随机控制理论，设计可行有效的数值格式;（3）将理论成果应用于生物学、数理金融学和风险管理。该项目的预期结果将有助于深入了解广泛的一类复杂随机系统。这反过来又将促进这种系统在金融和生物等领域的应用。状态转换扩散的混合正则和奇异控制问题可能会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的新问题，本研究项目的动机是新兴的应用所产生的生态系统建模，金融工程，保险风险过程，制造和生产计划。这些系统的动态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）存在日常抖动，这会导致小的波动以及由罕见事件（如流行病，地震和海啸）引起的大的种群损失;以及（ii）由于许多物种的生长速度和承载能力往往因环境条件而异，营养、供水和/或食物资源的变化。在保险风险过程的动态、风险资产的价格等方面也可以观察到类似的现象。这些特征使得文献中的通常模型不足以描述这样复杂的系统。拟议的项目旨在考虑到这些固有的随机力，并提出随机过程和相关的控制问题，这些问题是通用的，灵活的，但数学上易于处理，在处理这些现实世界的应用。它提出了新的随机过程的建模和分析的复杂系统，获得长时间的行为，这样的系统，发展奇异控制理论，并设计数值方案的控制问题。学生培训和教育，学科和跨学科的合作，以及通过出版物和演示文稿的研究成果的传播是这个项目的组成部分。