Stochastic volatility and fractional Brownian motion

AbstractWe observe (Yt) at times i/n, i=0,…,n, in the parametric stochastic volatility modeldYt=Φ(θ,WtH)dWt,where (Wt) is a Brownian motion, independent of the fractional Brownian motion (WtH) with Hurst parameter H⩾12. The sample size n increases not because of a longer observation period, but rather, because of more frequent observations.We prove that the unusual rate n−1/(4H+2) is asymptotically optimal for estimating the one-dimensional parameter θ, and we construct a contrast estimator based on an approximation of a suitably normalized quadratic variation that achieves the optimal rate