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

TL;DR

This paper presents a comprehensive error analysis framework for learning fractional stochastic differential equations from data, incorporating neural network-based coefficient estimation and validated through numerical experiments.

Contribution

It introduces a unified error analysis framework for fractional SDEs and demonstrates neural network methods for coefficient estimation within this context.

Findings

Derived convergence rates considering trajectory regularity

Identified main error sources in fractional SDE fitting

Validated theoretical results with numerical experiments

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.