Semiparametric estimation of volatility functions of diffusion processes from discretely observed data

This paper provides a semiparametric model to estimate processes of the volatility defined as the squared diffusion coefficient of a stochastic differential equation. Without assuming any functional form of the volatility function, we estimate the volatility process by filtering. We prove the consistency of the model in the sense that estimated processes converge to the true ones as the number of observations (N) goes to infinity and the sampling time interval (∆t) goes to zero while N∆t going to infinity. We also carry out numerical experiments through stochastic differential equations with linear/nonlinear volatility functions in order to check whether or not the model can actually estimate the volatility and compare the performance with the local linear model.