Yaozhong Hu: 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>0.5, it was known that the existing (naive) Euler scheme has the rate of convergence n^{1-2H}.

This scheme has a disadvantage, since when H converges to 0.5 (the standard Brownian motion case), there is no convergence!
Since the limit of the SDE as H goes to 0.5 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=0.5, the convergence rate of the naive Euler scheme deteriorates for H converging to 0.5. In this talk I will present a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for H=0.5 and it has the rate of convergence gamma_n^{-1}, where gamma_n=n^{ 2H-0.5} when H < 0.75, gamma_n= n/ \sqrt{ logn } when H = 0.75 and gamma_n=n if H> 0.75.

Furthermore,  I will also present the asymptotic behavior of the fluctuations of the error. More precisely, if {X_t, 0≤ t≤T} is the solution of a SDE driven by a fBm and if {X_t^n, 0≤ t≤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 is in (0.5, 0.75).

In the case H > 0.75, 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.