Topics in stochastic analysis and Malliavin calculus

Michigan State University, East Lansing MI

Investigators

Abstract

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.