Local Asymptotic Mixed Normality Property for Elliptic Diffusion: A Malliavin Calculus Approach

Abstract

We address the problem of the validity of the local asymptotic mixed normality (LAMN) property when the model is a multidimensional diffusion process X whose coefficients depend on a scalar parameter θ: the sample (Xk/n)0≤ k≤ n corresponds to an observation of X at equidistant times in the interval [0,1]. We prove that the LAMN property holds true for the likelihood under an ellipticity condition and some suitable smoothness assumptions on the coefficients of the stochastic differential equation. Our method is based on Malliavin calculus techniques: in particular, we derive for the log-likelihood ratio a tractable representation involving conditional expectations.