Penalized projection estimator for volatility density

In this paper, we consider a stochastic volatility model (Y-t, V-t), where the volatility (V-t) is a positive stationary Markov process. We assume that (InVt) admits a stationary density f that we want to estimate. Only the price process Y-t is observed at n discrete times with regular sampling interval Delta. We propose a non-parametric estimator for f obtained by a penalized projection method. Under mixing assumptions on (V-y), we derive bounds for the quadratic risk of the estimator. Assuming that Delta = Delta(n) tends to 0 while the number of observations and the length of the observation time tend to infinity, we discuss the rate of convergence of the risk. Examples of models included in this framework are given.