Extreme Value theory and Poisson statistics for discrete time samplings of stochastic differential equations

TL;DR

This paper explores the distribution of extreme events in stochastic processes derived from SDEs by applying extreme value theory and Poisson statistics, using transfer operators and spectral analysis.

Contribution

It introduces a novel approach combining SDE theory and functional analysis to study extreme events in discretely sampled stochastic processes.

Findings

Spectral properties of transfer operators are characterized.

Distribution of extreme events is linked to Poisson statistics.

Method bridges stochastic analysis and dynamical systems theory.

Abstract

We investigate the distribution and multiple occurrences of extreme events stochastic processes constructed by sampling the solution of a Stochastic Differential Equation on $\mathbb{R}^n$. We do so by studying the action of an annealead transfer operators on ad-hoc spaces of probability densities. The spectral properties of such operators are obtained by employing a mixture of techniques coming from SDE theory and a functional analytic approach to dynamical systems.