Aspects of Stochastic Differential Geometry in Function Space

函数空间中的随机微分几何方面

• 批准号: 9980209
• 负责人: Salah-Eldin Mohammed
• 金额: $ 6万
• 依托单位: Southern Illinois University at Carbondale
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9980209&HistoricalAwards=false
• 关键词: Aspects; Stochastic; Differential; Geometry; Function

This career advancement award is to enable the PI to study stochastic differential systems with memory, stochastic partial differential equations under geometric constraints, and their numerical aspects. Such models arise in engineering and physical applications. The award has several objectives:The first objective is to offer the PI the opportunity to acquire new expertise in differential geometry and numerical analysis. This activity will take the form of a study of the current literature on (stochastic) differential geometry, stochastic partial differential equations and stochastic numerics. The PI will consult with leading experts in these rapidly-developing areas. The second objective is to introduce new techniques from differential geometry and numerics in order to develop a theory of stochastic systems with memory and geometric constraints. In addition to its great relevance to physical applications, it is expected that the study of infinite-dimensional stochastic systems with constraints would lead to very interesting connections with current and fast-growing research in stochastic differential geometry. The third objective is introduce a dynamical and differential geometric component into the existing and fast-growing theory of stochastic partial differential equations. The PI will study the ergodic theory of stochastic flows generated by such systems. The fourth objective is to develop efficient algorithms for the numerical simulation of stochastic models with memory that arise in mechanical, control and electrical engineering, finance and economics, e.g. the option-pricing of financial securities whose evolution is governed by their past history.The fifth objective is to compile the results of the investigations into a graduate course targeting students majoring in mathematics, engineering and/or finance.

这个职业发展奖是为了使PI能够研究具有记忆的随机微分系统，几何约束下的随机偏微分方程及其数值方面。这种模型出现在工程和物理应用中。该奖项有几个目标：第一个目标是为PI提供获得微分几何和数值分析新专业知识的机会。这项活动将采取的形式是研究目前的文献（随机）微分几何，随机偏微分方程和随机数值。PI将咨询这些快速发展领域的领先专家。 第二个目标是引进新的技术，从微分几何和数值，以发展理论的随机系统的记忆和几何约束。除了与物理应用的密切相关性之外，人们还期望对具有约束的无限维随机系统的研究将导致与当前和快速增长的随机微分几何研究的非常有趣的联系。第三个目标是将动力学和微分几何引入到现有的和快速发展的随机偏微分方程理论中。PI将研究由这种系统产生的随机流的遍历理论。 第四个目标是为机械、控制和电气工程、金融和经济中出现的具有记忆的随机模型的数值模拟开发有效的算法，例如金融证券的期权定价，其演变受其过去历史的支配。第五个目标是将调查结果汇编成针对数学专业学生的研究生课程，工程和/或金融。