LAN property for ergodic diffusions with discrete observations

Abstract

We consider a multidimensional elliptic diffusion X α, β , whose drift b( α, x) and diffusion coefficients S( β, x) depend on multidimensional parameters α and β. We assume some various hypotheses on b and S, which ensure that X α, β is ergodic, and we address the problem of the validity of the Local Asymptotic Normality (LAN in short) property for the likelihoods, when the sample is ( X kΔ n ) 0⩽ k⩽ n , under the conditions Δ n →0 and nΔ n →+∞. We prove that the LAN property is satisfied, at rate nΔ n for α and n for β: our approach is based on a Malliavin calculus transformation of the likelihoods.