Some bounds for the maximum of a fractional Brownian motion

We'll consider bounds for the expectation of the maximum of a fractional Brownian motion with Hurst parameter $H$ and its approximations by discrete-time processes. The main result shows that the difference of the expectations for the continuous-time process and a discrete approximation in n points can be estimated from above by a quantity of order $\sqrt{\log n}/n^H$. We'll also give a simple proof of that when $H$ tends to zero, the expectation of the maximum of a fractional Brownian motion can be bounded from above and below by quantities of order $1/\sqrt{H}$.