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

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter H>1/2, it is known that the existing (naive) Euler scheme has the rate of convergence n1-2H. Since the limit H→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=1/2, the convergence rate of the naive Euler scheme deteriorates for H→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=1/2, and it has the rate of convergence γₙ⁻¹, where γₙ=n^2H-1/2 when H<3/4, γₙ=n/√log n when H=3/4 and γₙ=n if H>3/4. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if ₜ,0≤ t≤ T\ is the solution of a SDE driven by a fBm and if ₜⁿ,0≤ t≤ T\ is its approximation obtained by the new modified Euler scheme, then we prove that γₙ(Xⁿ-X) converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when H∈(1/2,3/4]. In the case H>3/4, we show the Lᵖ convergence of n(Xⁿₜ-Xₜ), 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.