Topics in stochastic analysis and Malliavin calculus

中心极限定理（CLT）是社会科学和自然科学中大量统计分析所依据的独立同分布试验的普适性结果。CLT的主要结论是，汇总数据遵循所谓的高斯定律，也称为正态或“钟形”曲线。但是，从地震学到计算机科学再到定量金融学的许多领域的科学家都发现，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对相关数据序列和相关问题的研究表明，CLT提供的高斯定律行为持续到很长的相关长度，与标准CLT有一些定量差异，例如平均值的分散程度增加。例如，PI的一个理论假设是，如果相关性足够长，那么实际上需要太多的数据才能观察到CLT类型的聚合。PI将研究更长范围的相关性的影响，表明数据可能涉及更高水平的不确定性，而不是钟形曲线行为。重尾），具有极慢的聚集速率。这在应用于住房市场的金融风险时可能具有一定的意义：可以为高度相关的抵押贷款的机构抵押贷款保险产品的卖方开发工具;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年世界金融危机前几年所犯的错误，这导致了纳税人资助的超过1800亿美元的救助。PI还计划研究所谓的自旋模型中的长程相关性的含义，这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，长程相关性和重尾对平均大尺度行为的影响很小或没有影响。私家侦探的博士学位。学生将参与研究的理论和应用方面，与PI一起证明定理，并使用数值在实践中测试他们的结果。让学生参与基础研究与现实世界的应用将广泛传播科学的理解。PI系统地鼓励来自代表性不足群体的学生加入研究计划。 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的变化的渐近规律的复杂性将通过搜索在一般情况下暗示正常，非正常和有条件正常极限的条件来证明，包括尖锐的收敛速度。第二，PI将分析密度，尾部，凸泛函，自旋系统和命中概率，一般Malliavin可微非高斯过程和领域。一套主要的工具是新使用的Malliavin演算的定量估计的各种距离之间的法律的随机变量的维纳空间。这包括PI的公式一般随机变量的密度在维纳空间，证明与I。2009年的努尔丁。另一个工具是PI对维纳空间上随机向量和场的凸泛函的比较，在2013年与I. Nourdin和G.佩卡蒂另一个是第一次精确估计Wiener空间上的正常定律的距离，由Bierme，Bonami，Nourdin和Peccati在2012年和2013年证明。PI将尽可能放弃幂尺度模型假设，如自相似性和/或平稳性，而是使用一般协方差结构固有的假设。这项工作的结果之一将是表明，众所周知的行为，在所谓的关键情况下，功率变化可以是所选择的模型类的文物。另一个目标是找出随机介质中自旋系统的所谓谢林顿-柯克帕特里克普适类的扩展，并确定当重尾和长程关联导致自旋系统退出该类时的行为。第三个结果应该是理解分数布朗运动的命中概率的临界情况。

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.