  We propose a new statistical observation scheme of diffusion processes named
convolutional observation, where it is possible to deal with smoother
observation than ordinary diffusion processes by considering convolution of
diffusion processes and some kernel functions with respect to time parameter.
We discuss the estimation and test theories for the parameter determining the
smoothness of the observation, as well as the least-square-type estimation for
the parameters in the diffusion coefficient and the drift one of the latent
diffusion process. In addition to the theoretical discussion, we also examine
the performance of the estimation and the test with computational simulation,
and show an example of real data analysis for one EEG data whose observation
can be regarded as smoother one than ordinary diffusion processes with
statistical significance.
