Error analysis for learning fractional stochastic differential equations with applications in neural approximations

Abstract

This paper develops a framework for the error analysis in nonparametric model fitting of fractional stochastic differential equations based on discrete observations. We identify and quantify the main error sources -- time discretization, coefficient approximation, and model fitting error -- within a unified framework. Through Sobolev-type norms, we derive convergence rates that incorporate the regularity of trajectories, thereby capturing the interaction of these error components. To demonstrate the applicability of the theory, we introduce a training scheme for coefficient function estimation based on shallow neural networks and a recurrent architecture. Numerical experiments validate the theoretical findings and illustrate the effectiveness of the approach.