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

Yanghui Liu, Samy Tindel|arXiv (Cornell University)|Mar 10, 2017

Stochastic processes and financial applications4 citations

TL;DR

This paper introduces and analyzes a modified first-order Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with Hurst parameter $ H \in (\frac{1}{3}, \frac{1}{2}) $, a regime known as the 'rough' case. By lifting the scheme and its error process to a rough path framework, the authors establish a convergence rate of order $ n^{\frac{1}{2}-2H} $, prove the rate is sharp, and derive a central limit theorem for the renormalized error using novel asymptotic techniques for weighted random sums.

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

Motivation & Objective

• To develop a numerically implementable first-order time-discrete scheme for SDEs driven by fractional Brownian motion in the rough regime $ H \in (\frac{1}{3}, \frac{1}{2}) $.
• To establish a convergence rate for the modified Euler scheme in this rough case, where classical Euler schemes diverge.
• To prove that the convergence rate $ n^{\frac{1}{2}-2H} $ is sharp and exact for the scheme.
• To derive a central limit theorem for the renormalized error process using advanced asymptotic analysis of weighted random sums.
• To demonstrate that the triple of processes—fBm, scheme, and normalized error—can be lifted to a rough path with a Hölder norm independent of the time step size.

Proposed method

• The scheme is constructed as a first-order time-discrete approximation, generalizing the classical Euler scheme to the rough case $ H < \frac{1}{2} $.
• The method relies on lifting the fBm, the scheme process, and the normalized error process into a rough path framework to control the convergence behavior.
• A key technical innovation is the use of a discrete sewing lemma to control the error in the scheme's approximation.
• The authors analyze weighted random sums via rough path techniques to derive the asymptotic distribution of the error.
• The convergence rate is derived by estimating the Hölder norm of the lifted rough path, which is shown to be independent of the time step size $ h = T/n $.
• A central limit theorem for the renormalized error is established using the convergence of the lifted rough path and the structure of iterated integrals of fBm.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate for a first-order implementable numerical scheme in the rough SDE case with $ H \in (\frac{1}{3}, \frac{1}{2}) $?
• RQ2 Can the modified Euler scheme be rigorously analyzed and proven to converge in the rough path sense for $ H < \frac{1}{2} $?
• RQ3 Is the convergence rate $ n^{\frac{1}{2}-2H} $ sharp, and can it be proven to be exact?
• RQ4 What is the asymptotic distribution of the renormalized error in this scheme?
• RQ5 Can the scheme and its error be lifted to a rough path with a norm independent of the time step size?

Key findings

• The modified Euler scheme achieves a convergence rate of order $ n^{\frac{1}{2}-2H} $ for SDEs driven by fBm with $ \frac{1}{3} < H < \frac{1}{2} $.
• This convergence rate is proven to be exact, meaning no faster rate is possible under the given assumptions.
• A central limit theorem is established for the renormalized error process, providing a non-Gaussian limit distribution due to the rough path structure.
• The triple of processes—fBm, scheme, and normalized error—can be lifted to a rough path whose Hölder norm is uniformly bounded with respect to the time step size.
• The analysis relies on novel asymptotic techniques for weighted random sums, particularly through the use of the discrete sewing lemma and rough path lifting.
• The results extend the applicability of Euler-type schemes to the rough case, where classical schemes fail, and provide a rigorous foundation for numerical implementation.