Topics in stochastic analysis and Malliavin calculus

中心极限定理（CLT）是独立同分布试验的普适性结果，在社会学和自然科学的统计分析中占有重要地位。CLT的主要结论是，汇总数据遵循所谓的高斯定律，也称为正态曲线或“钟形”曲线。但是，从地震学到计算机科学再到定量金融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可能是也可能不是观察这些数据如何聚集的有效方法。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有一些定量差异，例如平均分布的增加。例如，PI的一个理论推测是，如果相关性足够长，在实践中需要太多的数据才能观察到clt类型的聚集。PI将研究更长期相关性的影响，表明数据可能包含更高水平的不确定性（又名重尾），聚集速度极慢，而不是钟形曲线行为。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，比如美国国际集团（AIG）在2008年世界金融危机爆发前的几年里所犯的错误，那次危机导致纳税人支付了超过1800亿美元的救助资金。PI还计划研究所谓的自旋模型中远程相关性的含义，该模型在随机介质的物理学中很有用，与基于抵押贷款的金融衍生品的例子不同，在随机介质中，远程相关性和重尾对平均大规模行为的影响很小或没有影响。PI的博士生将参与理论和应用方面的研究，与PI一起证明定理，并在实践中使用数字测试他们的结果。让学生参与具有实际应用的基础研究将广泛传播科学认识。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提出了一项为期三年的随机分析研究计划，包括两组主题。首先，对于具有长期相关性的高斯过程变化的渐近规律的复杂性将通过在一般情况下搜索暗示正态，非正态和条件正态极限的条件来证明，包括急剧收敛速率。其次，PI将分析密度、尾和凸泛函、自旋系统和命中概率，用于一般的malliavin可微非高斯过程和场。一套主要的工具是在维纳空间上对随机变量定律之间的各种距离进行定量估计的Malliavin演算的新应用。这包括由I. Nourdin在2009年证明的关于Wiener空间上一般随机变量密度的PI公式。另一个工具是PI对Wiener空间上随机向量和场的凸泛函的比较，这是由I. Nourdin和G. Peccati在2013年证明的。另一个是在2012年和2013年由Bierme， Bonami， Nourdin和Peccati证明的第一个对Wiener空间正常律距离的精确估计。PI将尽可能放弃功率尺度模型的假设，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。这项工作的一个结果将是表明，在所谓的权力变化的关键情况下，众所周知的行为可以是所选模型类的工件。另一个目标是找出随机介质中自旋系统的所谓谢林顿-柯克帕特里克普适性类的扩展，并确定当重尾和远程相关性导致自旋系统退出该类时的行为。第三个结果应该是理解达到分数布朗运动概率的临界情况。

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.