LAMN property for hidden processes: The case of integrated diffusions

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 ∫₀ ¹ X _{(s + i)/n} dμ(s) for i = 0,...,n - 1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither Markovian nor 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.