Stochastic volatility and fractional Brownian motion

We observe (Y-1) at times i/n, i = 0,..., n, in the parametric stochastic volatility model d Y-t = Phi(theta,W-t(H)) dW(t), where (W-t) is a Brownian motion, independent of the fractional Brownian motion (W-t(H)) with Hurst parameter H greater than or equal to 1/2. 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 theta, and we construct a contrast estimator based on an approximation of a suitably normalized quadratic variation that achieves the optimal rate. (C) 2004 Elsevier B.V. All rights reserved.