Extreme Value Theory for diffusive particle systems with mean-field interaction

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Classical Extreme Value Theory studies the behavior of the largest values in large collections of numerical observations of some random phenomenon, for example, the intensities of earthquakes that occurred during a certain period at a certain region on the Richter scale. Existing results allow for understanding the likelihood of observing extremely large values in the future, larger than any past observation, and thus they can help in the prediction, for example, of extreme natural phenomena. This project aims to extend this theory to sets of interdependent numerical values, with a certain kind of interaction that is present in models which are widely used in Finance, Medicine, and other areas. Models of this type are used to describe: (a) quantities like the default likelihoods of competing financial institutions, where an extreme value analysis can help in the estimation of the probability of a credit event in the future; (b) the market capitalizations of all the companies in some large market, where an understanding of the extreme order statistics can be helpful in the improvement of existing models in stochastic portfolio theory; (c) The electrical potentials in human neurons, where extreme values might be associated with certain diseases. This award will also provide opportunities for students to be involved in this research. The aim of this project is to study the convergence of the appropriately normalized upper and intermediate order statistics of certain systems of SDEs as their size grows towards infinity. The equations interact through the dependence of the coefficients either directly on the systemic empirical measure, or on control processes that are picked to minimize some costs which are functions of the empirical measure. The first step is the reduction to the case where the SDEs are independent through the establishment of propagation of chaos, while the second step involves the treatment of this simple case using techniques from classical Extreme Value Theory and Malliavin Calculus. As statistical estimators for the parameters of the limiting distributions are functions of intermediate order statistics, properties like estimator consistency will also be extended to the case of interacting diffusions, allowing for the estimation of extreme value parameters associated with observed populations. Then, the probability of observing very large values in the future can be estimated by performing a time series analysis on the estimated parameters. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.