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

This paper is concerned with the discrete time stochastic volatility model Y-i = exp(X-i/2)eta(i), Xi+1 = b(X-i) + sigma(X-i)xi(i+1), where only (Y-i) is observed. The model is rewritten as a particular hidden model: Z(i) = X-i + epsilon(i), Xi+1 = b(X-i) + sigma(X-i)xi(i+1), where (xi(i)) and (epsilon(i)) are independent sequences of i.i.d. noise. Moreover, the sequences (X-i) 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 Z(1), ...., Z(n) are available. We propose to estimate bf and (b(2) + 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. (C) 2009 Elsevier B.V. All rights reserved.