Topics in stochastic analysis and Malliavin calculus

中心极限定理(CLT)是独立的、同分布的试验的普适性结果，其基础是社会学和自然科学中的许多统计分析。CLT的主要结论是，聚合数据遵循所谓的高斯定律，也就是众所周知的正态或“钟形”曲线。但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可能是也可能不是研究这些数据如何汇总的有效方式。PI在相关数据序列和相关问题上的工作将表明，CLT提供的高斯律行为持续到很长的相关长度，与标准CLT有一些数量上的差异，例如分布平均数的增加。例如，PI的一个理论猜想是，如果关联足够长，实际上需要太多的数据才能观察到CLT类型的聚集。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(又名。厚重的尾巴)，聚集速度极慢。这在应用于房地产市场的金融风险时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几年所做的错误，导致纳税人出资救助高达1800亿美元。PI还计划研究在随机介质物理学中有用的所谓自旋模型中的长期相关性的含义，在这种模型中，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对平均大规模行为几乎没有影响。PI的博士生将参与这项研究的理论和应用方面的研究，与PI一起证明定理，并在实践中使用数值来检验他们的结果。让学生参与具有现实世界应用的基础研究将广泛传播科学理解。国际学生联合会系统地鼓励来自代表性不足群体的学生加入研究计划。国际和平研究所提出了一个为期三年的随机分析研究计划，包括两组主题。首先，具有长程相关性的高斯过程变化的渐近律的复杂性将通过在一般情况下搜索暗示正常、非正常和条件正常极限的条件来证明，包括急剧的收敛速度。其次，PI将分析一般Malliavin可微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。一套主要的工具是Malliavin演算的新用途，用于定量估计Wiener空间上随机变量定律之间的各种距离。这包括关于Wiener空间上一般随机变量密度的PI公式，该公式由I.Nourdin在2009年证明。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，这是在2013年与I.Nourdin和G.Peccati一起证明的。另一个是Wiener空间上距离正常定律的第一个精确估计，Bierme，Bonami，Nourdin和Peccati在2012年和2013年证明了这一估计。PI将尽可能放弃自相似性和/或平稳性等幂尺度模型假设，转而使用一般协方差结构所固有的假设。这项工作的结果之一将是表明，在所谓的功率变化的关键情况下，众所周知的行为可能是选定的模型类的产物。另一个将是找出随机介质中自旋系统的所谓Sherrington-Kirkpatrick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这一类时的行为。第三个结果应该是理解分数布朗运动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.