Rates of convergence of a binary classification procedure for time-homogeneous S.D.E paths

TL;DR

This work studies binary classification of trajectories from time-homogeneous SDE mixtures, where both the drift and diffusion are unknown. It develops a nonparametric plug-in classifier built from spline-based drift estimators and a nonparametric diffusion estimator, and analyzes its minimax excess-risk rate over Hölder function classes. The main results establish upper and lower bounds: in general, the excess risk scales as $\exp\left(c\sqrt{\log N}\right)N^{-\beta/(2\beta+1)}$, with a matching lower bound $cN^{-\beta/(2\beta+1)}$, while a special diffusion-constant model yields the faster $N^{-1/2}$ rate (up to log factors) when $n \propto N$, and a known diffusion coefficient tightens the bound. The paper also extends drift estimation on the real line, uses a B-spline projection approach with growing truncation, and provides proofs leveraging transition-density representations and Girsanov changes of measure, offering practical guidance for diffusion-path-based classification in nonparametric settings.

Abstract

In the context of binary classification of trajectories generated by time-homogeneous stochastic differential equations, we consider a mixture model of two diffusion processes characterized by a stochastic differential equation whose drift coefficient depends on the class or label, which is modeled as a discrete random variable taking two possible values and whose diffusion coefficient is independent of the class. We assume that the drift and diffusion coefficients are unknown as well as the law of the discrete random variable that models the class. In this paper, we study the minimax convergence rate of the resulting nonparametric plug-in classifier under different sets of assumptions on the mixture model considered. As the plug-in classifier is based on nonparametric estimators of the coefficients of the mixture model, we also study a minimax convergence rate of the risk of estimation of the drift coefficients on the real line.

Theorems & Definitions (14)

• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Proposition 3.6
• Theorem 3.7
• proof
• proof
• Lemma 5.1
• proof