  This paper provides a semiparametric model of estimating states of the
volatility defined as the squared diffusion coefficient of a stochastic
differential equation. Without assuming any functional form of the volatility
function, we consider the volatility state as an unobservable state in a state
space model and estimate it by filtering. By using the estimates, we can draw
the information on what kind of functions the volatility has behind the
observable state of the process. We also prove the consistency of the model in
the sense that estimated states converges to the true ones as the observation
time interval goes to zero. In addition to that, from a numerical point of
views, we carry out numerical experiments by examples of stochastic
differential equations with linear/nonlinear volatility functions in order to
check whether or not the model can actually estimate the volatility and capture
the information on its functional form.
