Density and tail estimates via Malliavin calculus, and applications

Purdue University, West Lafayette IN

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI's three-year research program will investigate fundamental aspects of random variables which can be understood within the framework of Wiener spaces. Specifically, in the context of the Wiener process W (standard Brownian motion), if a random variable X can be written as a function of the path of W which is differentiable in the sense of Malliavin, meaning that its Frechet derivative DX in the direction of any appropriate perturbation exists, then it is possible to form a function g, equal to an averaged inner product of DX and of an exponentially correlated copy of DX, and use this function g to write estimates for the tails and even the density of X. An indication of this methodology is recorded in an article by the PI and Ivan Nourdin. The PI plan to apply the methodology to find sharp upper and lower bounds on densities of random variables of interest to probabilists, including the maxima of Gaussian fields, and also to tackle related problems such as small ball probabilities for fractional Brownian motion. A connection between Malliavin derivatives and Stein's method, which was discovered by Nourdin and Peccati, will also be investigated, and may help in analyzing random variables whose behavior is closer to non Gaussian distributions, including Gamma distributions, within the so-called Pearson class. The broader scientific significance of the proposed research begins with applications to the effect of chaotic environments on the stabilization or destabilization of physical or chemical systems, including polymers in random media. There should be a range of spatial correlation lengths in the medium which imply a continuum of behaviors, exhibiting richer phenomena than what theoretical physicists have predicted. Taking the modeling further, the PI plans to analyze the practical consequences of the project in those areas where long memory is an empirical fact, including financial econometrics, internet traffic, and climate prediction. Ph.D. students will take part in the fundamental aspects of the research. Some theoretical quantitative issues, such as small ball constants, fluctuation exponents, and long-memory parameter estimation, will be complemented with numerical simulations conducted by MS and undergraduate students. Involving students in fundamental research with real-world applications will broadly disseminate scientific understanding. The PI will encourage students from underrepresented groups to join this research program.