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April 2016 Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions
Yaozhong Hu, Yanghui Liu, David Nualart
Ann. Appl. Probab. 26(2): 1147-1207 (April 2016). DOI: 10.1214/15-AAP1114

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.

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Yaozhong Hu. Yanghui Liu. David Nualart. "Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions." Ann. Appl. Probab. 26 (2) 1147 - 1207, April 2016. https://doi.org/10.1214/15-AAP1114

Information

Received: 1 August 2014; Published: April 2016
First available in Project Euclid: 22 March 2016

zbMATH: 1339.60095
MathSciNet: MR3476635
Digital Object Identifier: 10.1214/15-AAP1114

Subjects:
Primary: 60H10
Secondary: 26A33 , 60H07 , 60H35

Keywords: Euler scheme , Fourth moment theorem , fractional Brownian motion , Fractional calculus , Malliavin calculus , Stochastic differential equations

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 2 • April 2016
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