Estimation of the volatility persistence in a discretely observed diffusion model

We consider the stochastic volatility model d Y t = σ t d B t , with B a Brownian motion and σ of the form σ t = Φ ( ∫ 0 t a ( t , u ) d W u H + f ( t ) ξ 0 ) , where W H is a fractional Brownian motion, independent of the driving Brownian motion B , with Hurst parameter H ≥ 1 / 2 . This model allows for persistence in the volatility σ . The parameter of interest is H . The functions Φ , a and f are treated as nuisance parameters and ξ 0 is a random initial condition. For a fixed objective time T , we construct from discrete data Y i / n , i = 0 , … , n T , a wavelet based estimator of H , inspired by adaptive estimation of quadratic functionals. We show that the accuracy of our estimator is n − 1 / ( 4 H + 2 ) and that this rate is optimal in a minimax sense.