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.

中央限制定理（CLT）是独立和分布的试验的普遍性结果，在社会学和自然科学中基于许多统计分析。 CLT的主要结论是，汇总数据遵循所谓的高斯定律，也称为正常或“铃”曲线。但是，从地震学到计算机科学再到定量金融的许多领域的科学家都发现，他们的数据系列具有长期相关性，这意味着CLT可能是看待此类数据聚合方式的有效方法。 PI在相关的数据序列和相关问题上的工作将表明，CLT提供的高斯法律行为持续到很长的相关长度，并且与标准CLT有一些定量差异，例如增加分散平均值的增加。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，无法观察CLT型聚合。 PI将研究甚至更长相关的效果，表明数据可能涉及更高水平的不确定性（又称重型尾巴），并且聚集速度极慢。当住房市场中应用于财务风险时，这可能具有一定意义：可以为机构抵押保险产品的卖方开发工具，以高度相关的抵押贷款；它们将有助于避免风险计算中的错误，例如在2008年世界金融危机之前的几年中，美国国际集团（AIG）犯了错误，这导致了纳税人资助的救助额超过1800亿美元。 PI还计划研究所谓的自旋模型中远程相关性的含义，这些模型在随机媒体的物理学中很有用，与基于抵押的金融衍生物的示例不同，远程相关性和重型尾巴可能对平均大规模行为的影响很小或没有影响。 PI的博士学位学生将参与研究的理论和应用方面，与PI合作以证明定理并使用数字测试其实践结果。让学生参与现实世界应用的基础研究将广泛传播科学理解。 PI系统地鼓励来自代表性不足的小组的学生加入研究计划。 PI提出了一项为期三年的随机分析研究计划，并提出了两组主题。首先，通过寻找条件，这意味着在一般情况下，包括正常，非正态和有条件正常的限制，包括急剧的收敛率，可以证明渐近定律对具有远距离相关性的高斯过程变化的复杂性。其次，PI将分析一般的Malliavin差异性非高斯工艺和领域，分析密度，尾部和凸功能，自旋系统和击中概率。一组主要的工具是Malliavin演算的新使用，用于定量估计Wiener空间上随机变量定律之间的各种距离。这包括Wiener空间上一般随机变量密度密度的PI公式，该公式在2009年与I. Nourdin证明。另一个工具是PI在Wiener Space上随机向量和田地的凸功能的比较，并于2013年与I. Nourdin和G. Peccati进行了证明。另一个是对维也纳空间正常法律距离的第一个尖锐估计，该法律在2012年和2013年由Bierme，Bonami，Nourdin和Peccati证明。 PI将在可能的情况下放弃诸如自相似性和/或平稳性之类的功率尺度模型假设，而是使用对一般协方差结构固有的假设。这项工作的结果之一是表明，所谓的关键案例中众所周知的行为可以是所选模型类的人工制品。另一个将是找出随机媒体中自旋系统的所谓Sherrington-Kirkpatrick通用类的扩展，并确定何时重尾巴和远程相关性导致旋转系统退出此类阶级。第三个结果应该是了解击中分数布朗运动概率的关键案例。