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.