Lan property for ergodic diffusions with discrete observations

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_kΔn)_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.