Robust Functional Data Analysis for Stochastic Evolution Equations in Infinite Dimensions

This article addresses the robust measurement of covariations in the context of solutions to stochastic evolution equations in Hilbert spaces using functional data analysis. For such equations, standard techniques for functional data based on cross-sectional covariances are often inadequate for identifying statistically relevant random drivers and detecting outliers since they overlook the interplay between cross-sectional and temporal structures. Therefore, we develop an estimation theory for the continuous quadratic covariation of the latent random driver of the equation instead of a static covariance of the observable solution process. We derive identifiability results under weak conditions, establish rates of convergence and a central limit theorem based on infill asymptotics, and provide long-time asymptotics for estimation of a static covariation of the latent driver. Applied to term structure data, our approach uncovers a fundamental alignment with scaling limits of covariations of specific short-term trading strategies, and an empirical study detects several jumps and indicates high-dimensional and time-varying covariations.