Fast rates for the classification of Ornstein-Uhlenbeck paths

Abstract

In binary classification problems, plug-in classifiers can sometimes achieve faster excess risk rates than the underlying estimation rate. In this work, we study a supervised classification problem based on repeated independent observations of Ornstein-Uhlenbeck paths, with both time-homogeneous and time-inhomogeneous drifts. We show that, under a suitable separation condition on the class-dependent drifts, a plug-in classifier attains an excess risk of quadratic order in the corresponding parametric or nonparametric estimation error. The theoretical approach utilizes the Wiener chaos representation and spectral theory to characterize the log-likelihood ratio as a quadratic form of Gaussian random variables, enabling a precise analysis of margin behaviour and deviation results. This extends the fast-rate results from classification problems with linear discriminant models and Gaussian white noise models to dynamical diffusion systems with Gaussian structure.