Strong convergence rate of the Euler scheme for SDEs driven by additive rough fractional noises

TL;DR

This paper establishes the strong convergence rates of the Euler scheme for SDEs driven by additive fractional Brownian motions with low Hurst parameters, using advanced mathematical tools to handle irregularities and unbounded drifts.

Contribution

It provides new convergence rate results for Euler schemes applied to SDEs with rough fractional noises and unbounded drifts, extending previous understanding.

Findings

Euler scheme has strong order 2H for bounded derivatives up to third order.

Strong order H+1/2 achieved for linear cases.

Numerical simulations confirm theoretical convergence rates.

Abstract

The strong convergence rate of the Euler scheme for SDEs driven by additive fractional Brownian motions is studied, where the fractional Brownian motion has Hurst parameter $H\in(\frac13,\frac12)$ and the drift coefficient is not required to be bounded. The Malliavin calculus, the rough path theory and the $2$D Young integral are utilized to overcome the difficulties caused by the low regularity of the fractional Brownian motion and the unboundedness of the drift coefficient. The Euler scheme is proved to have strong order $2H$ for the case that the drift coefficient has bounded derivatives up to order three and have strong order $H+\frac12$ for linear cases. Numerical simulations are presented to support the theoretical results.