Risk comparison theorems and application to deep learning of diffusion coefficients

Abstract

We investigate the nonparametric estimation of the diffusion matrix in stochastic differential equations featuring multidimensional, strong mixing covariate processes. We propose a flexible statistical framework based on general function classes that does not require a linear basis representation, rendering our results directly applicable to deep neural network estimators. Our approach employs a two-step estimation procedure: constructing a preliminary nonparametric quasi-likelihood estimator and subsequently regularizing it via a $\beta$-H\"older class approximation. We establish general risk comparison theorems between empirical and generalization risks for arbitrary estimators without relying on a specific probabilistic structure of the underlying process. In diffusion matrix learning based on $n + 1$ observations over the time interval $[0,T]$, the derived upper bounds capture the intrinsic interplay between the $T$-rate, associated with the mixing behavior of the covariate process, and the intrinsic $n$-rate governing the volatility estimation.