Nonparametric estimation for a stochastic volatility model

Consider discrete time observations (X_{\ell\delta})_{1\leq \ell \leq n+1}$of the process$X$satisfying$dX_t= \sqrt{V_t} dB_t$, with$V_t$a one-dimensional positive diffusion process independent of the Brownian motion$B$. For both the drift and the diffusion coefficient of the unobserved diffusion$V$, we propose nonparametric least square estimators, and provide bounds for theirrisk. Estimators are chosen among a collection of functions belonging to a finite dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.$