The Annals of Applied Probability

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

Yanghui Liu and Samy Tindel

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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 $\frac{1}{3}<H<\frac{1}{2}$. This is a first-order time-discrete numerical approximation scheme, and has been introduced in [Ann. Appl. Probab. 26 (2016) 1147–1207] recently in order to generalize the classical Euler scheme for Itô SDEs to the case $H>\frac{1}{2}$. The current contribution generalizes the modified Euler scheme to the rough case $\frac{1}{3}<H<\frac{1}{2}$. Namely, we show a convergence rate of order $n^{\frac{1}{2}-2H}$ 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 Hölder norm of this new rough path has an estimate which is independent of the step-size of the scheme.

Article information

Ann. Appl. Probab., Volume 29, Number 2 (2019), 758-826.

Received: March 2017
Revised: October 2017
First available in Project Euclid: 24 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H35: Computational methods for stochastic equations [See also 65C30] 65C30: Stochastic differential and integral equations

Rough paths discrete sewing lemma fractional Brownian motion stochastic differential equations Euler scheme asymptotic error distributions


Liu, Yanghui; Tindel, Samy. First-order Euler scheme for SDEs driven by fractional Brownian motions: The rough case. Ann. Appl. Probab. 29 (2019), no. 2, 758--826. doi:10.1214/17-AAP1374.

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