The Annals of Applied Probability

Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions

Yaozhong Hu, Yanghui Liu, and David Nualart

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For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for Itô SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $\gamma_{n}^{-1}$, where $\gamma_{n}=n^{2H-{1}/2}$ when $H<\frac{3}{4}$, $\gamma_{n}=n/\sqrt{\log n}$ when $H=\frac{3}{4}$ and $\gamma_{n}=n$ if $H>\frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_{t},0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_{t}^{n},0\le t\le T\}$ is its approximation obtained by the new modified Euler scheme, then we prove that $\gamma_{n}(X^{n}-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^{p}$ convergence of $n(X^{n}_{t}-X_{t})$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.

Article information

Ann. Appl. Probab., Volume 26, Number 2 (2016), 1147-1207.

Received: August 2014
First available in Project Euclid: 22 March 2016

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 26A33: Fractional derivatives and integrals 60H35: Computational methods for stochastic equations [See also 65C30]

Fractional Brownian motion stochastic differential equations Euler scheme fractional calculus Malliavin calculus fourth moment theorem


Hu, Yaozhong; Liu, Yanghui; Nualart, David. Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions. Ann. Appl. Probab. 26 (2016), no. 2, 1147--1207. doi:10.1214/15-AAP1114.

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