Error analysis for approximations to one-dimensional SDEs via the perturbation method

TL;DR

This paper analyzes the asymptotic error distributions of approximation schemes for one-dimensional SDEs driven by fractional Brownian motion, extending previous results to include drift terms and simplifying the proof process.

Contribution

It extends prior work on error analysis of fractional SDE approximations to cases with drift and provides a simplified proof method.

Findings

Derived the asymptotic error distribution as a directional derivative.

Extended error analysis to SDEs with drift terms.

Simplified the proof of remainder estimates.

Abstract

We study asymptotic error distributions associated with standard approximation scheme for one-dimensional stochastic differential equations driven by fractional Brownian motions. This problem was studied by, for instance, Gradinaru-Nourdin [6], Neuenkirch and Nourdin [14] and the second named author [13]. The aim of this paper is to extend their results to the case where the equations contain drift terms and simplify the proof of estimates of the remainder terms in [13]. To this end, we represent the approximation solution as the solution of the equation which is obtained by replacing the fractional Brownian path with a perturbed path. We obtain the asymptotic error distribution as a directional derivative of the solution by using this expression.