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>12, it is known that the existing (naive) Euler scheme has the rate of convergence n1−2H. Since the limit H→12 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=12, the convergence rate of the naive Euler scheme deteriorates for H→12. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for H=12, and it has the rate of convergence γ−1n, where γn=n2H−1/2 when H<34, γn=n/logn−−−−√ when H=34 and γn=n if H>34. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {Xt,0≤t≤T} is the solution of a SDE driven by a fBm and if {Xnt,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∈(12,34]. In the case H>34, 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.