Nonparametric estimation for a stochastic volatility model

Consider discrete-time observations (X-l delta) (1 <= l <= n + 1) of the process X satisfying dX(t) = root V(t)dB(t), with V a one-dimensional positive diffusion process independent of the Brownian motion B. For both the drift and the diffusion coefficient of the unobserved diffusion V, we propose nonparametric least square estimators, and provide bounds for their risk. Estimators are chosen among a collection of functions belonging to a finite-dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.