Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### LAMN property for elliptic diffusion: a Malliavin calculus approach

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Résumé: In this paper, 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 linear parameter $\theta$: the sample $(X_{k/n})_{0\leq k\leq n}$ corresponds to an observation of $X$ at equidistant times of the interval $[0,1]$. We prove that LAMN property holds true for the likelihoods, 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.