First-order Euler scheme for SDEs driven by fractional Brownian motions: the rough case

Yanghui Liu; Samy Tindel

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arXiv · 1703.03625

First-order Euler scheme for SDEs driven by fractional Brownian motions: the rough case

Yanghui Liu·Samy Tindel

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Abstract

In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $\frac13 \frac12$. The current contribution generalizes the modified Euler scheme to the rough case $\frac13<H<\frac12$. Namely, we show a convergence rate of order $n^{\frac12-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the H\"older norm of this new rough path has an estimate which is independent of the step-size of the scheme.

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BibTeX RIS

Yanghui Liu, Samy Tindel. 2017-03-10. First-order Euler scheme for SDEs driven by fractional Brownian motions: the rough case. https://arxiv.org/abs/1703.03625

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