Adaptive estimation of the dynamics of a discrete time stochastic volatility model

This paper is concerned with the discrete time stochastic volatility model Yi=exp(Xi/2)[eta]i, Xi+1=b(Xi)+[sigma](Xi)[xi]i+1, where only (Yi) is observed. The model is rewritten as a particular hidden model: Zi=Xi+[epsilon]i, Xi+1=b(Xi)+[sigma](Xi)[xi]i+1, where ([xi]i) and ([epsilon]i) are independent sequences of i.i.d. noise. Moreover, the sequences (Xi) and ([epsilon]i) are independent and the distribution of [epsilon] is known. Then, our aim is to estimate the functions b and [sigma]2 when only observations Z1,...,Zn are available. We propose to estimate bf and (b2+[sigma]2)f and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of b and [sigma]2 are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples.