Some related problems for stochastic partial differential equations and backward stochastic differential equations

随机偏微分方程和倒向随机微分方程的一些相关问题

• 批准号: 0906907
• 负责人: Jie Xiong
• 金额: $ 11.73万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906907&HistoricalAwards=false
• 关键词: Some; related; problems; stochastic; partial

The PI will study three types of problems about stochastic partial differential equations (SPDEs) and backward stochastic differential equations (BSDEs). Firstly, he will build a connection between SPDEs and BSDEs; and will use it to prove the strong uniqueness and regularity properties for the solutions to such SPDEs. He will also study the rare eventsphenomena for financial market using large deviation principle.Secondly, the PI will derive a stochastic maximum principle (SMP) for stochastic optimal control problems arising from real world applications where the coefficients of the systems are not smooth. This will lead to two new classes of BSDEs. He will study their solutions and will explore the numerical approximation for these. Finally,the PI will consider the parameter estimationproblem based on partial information. The key to the proof of theconsistency of such estimates is heavily dependent on theestablishment of a filtering stability result.This research is motivated from the study of certain populationmodels, the wireless communication and the mathematical finance. The PI will develop mathematic tools to study SPDEs and BSDEs arising from these problems. The results obtained will be applied to various real world applications for the benefitto the society. For example, the SMP can be applied to the power controlproblem to serve the society in improving the efficiency of thecommunication network. The study of the financial market will helpthe government to regulate the market; especially, the largedeviation problem will help to characterizerare events in the financial market. The parameter estimationproblem can be used to locate various targets including the hiddensource of pollution. Long-term effects on graduate and post-graduate training of students in stochastic analysis and optimal control are also expected.

PI将研究三种类型的随机偏微分方程（SPDE）和倒向随机微分方程（BSDEs）问题。 首先，他将建立SPDE和BSDES之间的联系;并将用它来证明这种SPDE的解的强唯一性和正则性。其次，PI将推导出一个随机最大值原理（SMP），用于解决由真实的系统系数不光滑的随机最优控制问题。这将导致两类新的BSDES。他将研究他们的解决方案，并将探讨这些数值近似。最后，PI将考虑基于部分信息的参数估计问题。证明这类估计的一致性的关键是建立一个滤波稳定性的结果，本研究的动机来自于对某些人口模型、无线通信和数学金融的研究。PI将开发数学工具来研究由这些问题引起的SPDE和BSDE。所获得的结果将应用于各种真实的世界的应用，造福社会。例如，SMP可以应用于功率控制问题，以服务于社会，提高通信网络的效率。对金融市场的研究将有助于政府对市场的调控，特别是大偏差问题的研究将有助于刻画金融市场中的一些事件。参数估计问题可以用来定位包括隐藏污染源在内的各种目标。还预计对随机分析和最佳控制专业的毕业生和研究生培训产生长期影响。