Estimation of the volatility persistence in a discretely observed diffusion model

We consider the stochastic volatility model dYt=σtdBt, with BB a Brownian motion and σσ of the form σt=Φ(∫0ta(t,u)dWuH+f(t)ξ0), where WHWH is a fractional Brownian motion, independent of the driving Brownian motion BB, with Hurst parameter H≥1/2H≥1/2. This model allows for persistence in the volatility σσ. The parameter of interest is HH. The functions ΦΦ, aa and ff are treated as nuisance parameters and ξ0ξ0 is a random initial condition. For a fixed objective time TT, we construct from discrete data Yi/n,i=0,…,nTYi/n,i=0,…,nT, a wavelet based estimator of HH, inspired by adaptive estimation of quadratic functionals. We show that the accuracy of our estimator is n−1/(4H+2)n−1/(4H+2) and that this rate is optimal in a minimax sense.