Stochastic differential equations: potential theory and uniqueness

随机微分方程：势论和唯一性

• 批准号: 0901505
• 负责人: Richard Bass
• 金额: $ 36万
• 依托单位: University of Connecticut
• 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=0901505&HistoricalAwards=false
• 关键词: Stochastic; differential; equations; potential; theory

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The behavior of harmonic functions and potential functions forprocesses that have jumps will be studied. In particular, how does smoothness in the coefficients of the operators translate tothe smoothness of harmonic and potential functions?Secondly, uniqueness for several probability models that have origins inmathematical physics and mathematical biology will be investigated. One model is that of stochastic partial differential equations in one dimension with a space-time white noise as a driving term. A second model is that of superprocesses with branching interactions.In recent years researchers in mathematical physics, economics, andmathematical finance have realized that to adequately model real-worldphenomena, the possibility of jumps must be allowed. For example, anunexpected discovery or unexpected regional conflict might cause a sudden jumpup or down in stock prices. However, some of the very basic questionsabout models that incorporate jumps are as yet unanswered. One of the areasto be investigated is the regularity of such models. A typical question is:if the initial data is perturbed slightly, is it true that the behavior ofthe model at future times will also be only slightly perturbed?

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。将研究具有跳跃的过程的调和函数和势函数的行为。特别是，如何平滑的系数的运营商转化为平滑的谐波和潜在的功能？其次，研究了数学物理学和数学生物学中的几个概率模型的唯一性问题。一种模型是一维随机偏微分方程，以时空白色噪声作为驱动项。第二个模型是具有分支相互作用的超过程模型。近年来，数学物理学、经济学和数理金融学的研究人员已经认识到，要充分模拟真实世界的现象，必须允许跳跃的可能性。例如，一个意想不到的发现或意想不到的地区冲突可能会导致股票价格突然上涨或下跌。然而，一些关于包含跳跃的模型的非常基本的问题还没有答案。其中一个有待研究的领域是这些模型的规律性。一个典型的问题是：如果初始数据受到轻微的扰动，那么模型在未来的行为是否也只会受到轻微的扰动？