LAMN property for hidden processes: the case of integrated diffusions

Abstract

In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process $X$. Our data are given by $ \int_0^1 X_{\frac{s+i}{n}} \dd \mu (s)$ for $i=0,...,n-1$ and the unknown parameter appears in the diffusion coefficient of the process $X$ only. Although the data are nor Markovian neither Gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for a usual discrete sampling of $X$.