Estimating the Diffusion Coefficient for Diffusions Driven by fBm

When the Hurst coefficient of a fBm B_t^H is greater than 1/2, it is possible to define a stochastic integral with respect to B_t^H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type X_t = x₀ + ₀^T (X_s) dB_s^H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X_t. We then study consistency in probability of this estimator and calculate convergence rates in probability.