Topics in stochastic analysis and Malliavin calculus

随机分析和 Malliavin 微积分主题

• 批准号: 1734183
• 负责人: Frederi Viens
• 金额: $ 5.55万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-07 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1734183&HistoricalAwards=false
• 关键词: Topics; stochastic; analysis; Malliavin; calculus

The central limit theorem (CLT) is a universality result for independent and identically distributed trials on which is based much statistical analysis in the sociological and natural sciences. The CLT's main conclusion is that aggregated data follows the so-called Gaussian law, also known as the normal or "bell" curve. But scientists in many fields from seismology to computer science to quantitative finance are finding that their data series have long-range correlations, which means that the CLT may or may not be a valid way of looking at how such data aggregates. The PI's work on correlated data sequences, and related questions, would show that the Gaussian-law behavior afforded by the CLT persists up to very long correlation lengths, with some quantitative differences with the standard CLT, such as an increase in how spread out averages tend to get. For instance, one of the PI's theoretical conjectures is that if correlation is long enough, it would take too much data in practice to be able to observe a CLT-type aggregation. The PI will study the effect of even longer-range correlations, showing that instead of bell-curve behavior, data could involve much higher levels of uncertainty (a.k.a. heavy tails), with an extremely slow rate of aggregation. This could be of some significance when applied to financial risk in the housing market: tools could be developed for sellers of institutional mortgage insurance products for highly correlated mortgages; they would help avoid errors in risk calculations, such as those made by the American International Group (AIG) in the years preceding the world financial crisis of 2008, which resulted in a taxpayer-funded bailout upwards of $ 180 billion. The PI also plans to study the implications of long-range correlations in so-called spin models which are useful in the physics of random media, where, unlike the example of mortgage-based financial derivatives, long-range correlations and heavy tails could have little or no influence on the average large-scale behavior. The PI's Ph.D. students will take part in both theoretical and applied aspects of the research, working with the PI to prove theorems and test their results in practice using numerics. Involving students in fundamental research with real-world applications will broadly disseminate scientific understanding. The PI systematically encourages students from underrepresented groups to join the research program. The PI proposes a three-year research program in stochastic analysis, with two groups of topics. First, the complexity of asymptotic laws for variations of Gaussian processes with long-range correlations will be evidenced by searching for conditions implying normal, non-normal, and conditionally normal limits in general situations, including sharp convergence rates. Second, the PI will analyze densities, tails, and convex functionals, spin systems, and hitting probabilities, for general Malliavin-differentiable non-Gaussian processes and fields. A main set of tools is the new use of the Malliavin calculus for quantitative estimates of various distances between laws of random variables on Wiener space. This includes the PI's formula for the density of general random variables on Wiener space, proved with I. Nourdin in 2009. Another tool is the PI's comparison of convex functionals for random vectors and fields on Wiener space, proved in 2013 with I. Nourdin and G. Peccati. Yet another is the first sharp estimates of distances to the normal law on Wiener space, proved in 2012 and 2013 by Bierme, Bonami, Nourdin, and Peccati. The PI will forego power-scale model assumptions such as self-similarity and/or stationarity whenever possible, using instead assumptions which are intrinsic to general covariance structures. One of the consequence of the work will be to show that well-known behaviors in so-called critical cases for power variations can be artefacts of the chosen model classes. Another will be to find out the extend of the so-called Sherrington-Kirkpatrick universality class for spin systems in random media, and to determine behaviors when heavy tails and long-range correlations cause spin systems to exit this class. A third consequence should be to understand the critical cases for hitting probabilities of fractional Brownian motion.

中心极限定理（CLT）是独立同分布试验的普遍性结果，其基础是社会学和自然科学中的大量统计分析。 CLT 的主要结论是聚合数据遵循所谓的高斯定律，也称为正态曲线或“钟形”曲线。但从地震学到计算机科学再到定量金融学等许多领域的科学家发现，他们的数据序列具有长期相关性，这意味着 CLT 可能是也可能不是研究此类数据聚合方式的有效方法。 PI 在相关数据序列和相关问题上的工作将表明，CLT 提供的高斯定律行为持续到非常长的相关长度，与标准 CLT 存在一些定量差异，例如平均分布方式的增加倾向于得到。例如，PI 的理论猜想之一是，如果相关性足够长，则在实践中需要太多数据才能观察到 CLT 类型的聚合。 PI 将研究更长期相关性的影响，表明数据可能涉及更高水平的不确定性（又称重尾），而不是钟形曲线行为，并且聚合速度极慢。当应用于房地产市场的金融风险时，这可能具有一定的意义：可以为高度相关抵押贷款的机构抵押贷款保险产品的卖家开发工具；它们将有助于避免风险计算中的错误，例如美国国际集团 (AIG) 在 2008 年世界金融危机之前的几年中所犯的错误，导致纳税人资助的救助金额高达 1800 亿美元。 PI 还计划研究所谓的自旋模型中的长期相关性的影响，该模型在随机媒体物理学中很有用，与基于抵押贷款的金融衍生品的例子不同，长期相关性和重尾可能会产生影响。对平均大规模行为影响很小或没有影响。 PI 的博士学位学生将参与理论和应用方面的研究，与 PI 合作证明定理并使用数字在实践中测试他们的结果。让学生参与基础研究和实际应用将广泛传播科学理解。 PI 系统地鼓励来自代表性不足群体的学生加入研究项目。 PI 提出了一个为期三年的随机分析研究计划，有两组主题。首先，具有长程相关性的高斯过程变化的渐近定律的复杂性将通过搜索一般情况下暗示正态、非正态和条件正态极限的条件（包括急剧收敛率）来证明。其次，PI 将分析一般 Malliavin 可微分非高斯过程和场的密度、尾部和凸泛函、自旋系统和命中概率。主要工具集是马利亚文微积分的新用途，用于定量估计维纳空间上随机变量定律之间的各种距离。这包括 PI 的维纳空间上一般随机变量密度公式，由 I. Nourdin 于 2009 年证明。另一个工具是 PI 对维纳空间上随机向量和场的凸函数的比较，于 2013 年由 I. Nourdin 证明， G.佩卡蒂。还有一个是对维纳空间正常定律距离的首次精确估计，由 Bierme、Bonami、Nourdin 和 Peccati 于 2012 年和 2013 年证明。只要有可能，PI 将放弃功率尺度模型假设，例如自相似性和/或平稳性，而使用一般协方差结构固有的假设。这项工作的结果之一将表明，所谓的功率变化关键情况下的众所周知的行为可能是所选模型类的假象。另一个目标是找出随机介质中自旋系统所谓的谢林顿-柯克帕特里克普遍性类的扩展，并确定当重尾和长程相关性导致自旋系统退出此类时的行为。第三个结果应该是了解分数布朗运动命中概率的关键情况。