Kernel deconvolution of stochastic volatility models

In this paper, we study the problem of the nonparametric estimation of the function m in a stochastic volatility model h(t) = exp(X-t/2lambda)xi(t), X-t = m(Xt-1) + eta(t), where xi(t) is a Gaussian white noise. We show that the model can be written as an autoregression with errors-in-variables. Then an adaptation of the deconvolution kernel estimator proposed by Fan and Truong [Annals of Statistics, 21, (1993) 1900] estimates the function m with the optimal rate, which depends on the distribution of the measurement error. The rates vary from powers of n to powers of ln(n) depending on the rate of decay near infinity of the characteristic function of this noise. The performance of the method are studied by some simulation experiments and some real data are also examined.