Density and tail estimates via Malliavin calculus, and applications

通过 Malliavin 演算进行密度和尾部估计以及应用

• 批准号: 0907321
• 负责人: Frederi Viens
• 金额: $ 23.07万
• 依托单位: Purdue University
• 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: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907321&HistoricalAwards=false
• 关键词: Density; tail; estimates; via; Malliavin

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The PI's three-year research program will investigate fundamental aspects of random variables which can be understood within the framework of Wiener spaces. Specifically, in the context of the Wiener process W (standard Brownian motion), if a random variable X can be written as a function of the path of W which is differentiable in the sense of Malliavin, meaning that its Frechet derivative DX in the direction of any appropriate perturbation exists, then it is possible to form a function g, equal to an averaged inner product of DX and of an exponentially correlated copy of DX, and use this function g to write estimates for the tails and even the density of X. An indication of this methodology is recorded in an article by the PI and Ivan Nourdin. The PI plan to apply the methodology to find sharp upper and lower bounds on densities of random variables of interest to probabilists, including the maxima of Gaussian fields, and also to tackle related problems such as small ball probabilities for fractional Brownian motion. A connection between Malliavin derivatives and Stein's method, which was discovered by Nourdin and Peccati, will also be investigated, and may help in analyzing random variables whose behavior is closer to non Gaussian distributions, including Gamma distributions, within the so-called Pearson class.The broader scientific significance of the proposed research begins with applications to the effect of chaotic environments on the stabilization or destabilization of physical or chemical systems, including polymers in random media. There should be a range of spatial correlation lengths in the medium which imply a continuum of behaviors, exhibiting richer phenomena than what theoretical physicists have predicted. Taking the modeling further, the PI plans to analyze the practical consequences of the project in those areas where long memory is an empirical fact, including financial econometrics, internet traffic, and climate prediction. Ph.D. students will take part in the fundamental aspects of the research. Some theoretical quantitative issues, such as small ball constants, fluctuation exponents, and long-memory parameter estimation, will be complemented with numerical simulations conducted by MS and undergraduate students. Involving students in fundamental research with real-world applications will broadly disseminate scientific understanding.The PI will encourage students from underrepresented groups to join this research program.

该奖项是根据2009年《美国回收与再投资法》（公法111-5）资助的。PI的三年研究计划将调查随机变量的基本方面，这些方面可以在Wiener空间的框架内理解。 Specifically, in the context of the Wiener process W (standard Brownian motion), if a random variable X can be written as a function of the path of W which is differentiable in the sense of Malliavin, meaning that its Frechet derivative DX in the direction of any appropriate perturbation exists, then it is possible to form a function g, equal to an averaged inner product of DX and of an exponentially correlated copy of DX, and use this function g to write PI和Ivan Nourdin的文章中记录了该方法的指示。 PI计划采用该方法，以发现对概率的随机变量的密度尖锐的上限和下限，包括高斯领域的最大值，并解决相关问题，例如小球概率的小球概率。 诺尔丁和佩卡蒂发现的malliavin衍生物与斯坦因的方法之间的联系也将进行研究，并可能有助于分析其行为更接近非高斯分布的随机变量，包括γ分布，包括粘液分布，在所谓的Pearson分类中，在拟议的稳定性方面开始了对CHA的更广泛的稳定性，从而对CHA的稳定性进行了更广泛的研究，从而使CHA的效率更加重要，从而实现了效应的效果。或化学系统，包括随机培养基中的聚合物。培养基中应该有一系列的空间相关长度，这意味着行为的连续性，表现出比理论物理学家所预测的更丰富的现象。进一步进行建模，PI计划在那些长期记忆是经验事实的领域中分析该项目的实际后果，包括金融计量经济学，互联网流量和气候预测。博士学生将参与研究的基本方面。一些理论定量问题，例如小球常数，波动指数和长期内存参数估计，将与MS和本科生进行的数值模拟相辅相成。让学生与现实世界应用一起参与基础研究将广泛传播科学理解。PI将鼓励来自代表性不足的团体的学生加入该研究计划。