The Annals of Applied Probability

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

Abstract

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

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

Dates
First available in Project Euclid: 22 March 2016

https://projecteuclid.org/euclid.aoap/1458651830

Digital Object Identifier
doi:10.1214/15-AAP1114

Mathematical Reviews number (MathSciNet)
MR3476635

Zentralblatt MATH identifier
1339.60095

Citation

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. https://projecteuclid.org/euclid.aoap/1458651830