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 ) η i , X i + 1 = b ( X i ) + σ ( X i ) ξ i + 1 , where only ( Y i ) is observed. The model is rewritten as a particular hidden model: Z i = X i + ε i , X i + 1 = b ( X i ) + σ ( X i ) ξ i + 1 , where ( ξ i ) and ( ε i ) are independent sequences of i.i.d. noise. Moreover, the sequences ( X i ) and ( ε i ) are independent and the distribution of ε is known. Then, our aim is to estimate the functions b and σ 2 when only observations Z 1 , … , Z n are available. We propose to estimate b f and ( b 2 + σ 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 σ 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.