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

### Parametric inference for discretely observed non-ergodic diffusions

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Résumé: We consider a multidimensional diffusion process $X$ whose drift and diffusion coefficients depend respectively on a parameter $\la$ and $\te$. This process is observed at $n+1$ equally--spaced times $0,\De_n,2\De_n,\ldots,n\De_n$, and $T_n=n\De_n$ denotes the length of the observation window''. We are interested in estimating $\la$ and/or $\te$. Under suitable smoothness and identifiability conditions, we exhibit estimators $\lan$ and $\ten$, such that the variables $\rn~(\ten-\te)$ and $\sqrt{T_n}~(\lan-\la)$ are tight, as soon as $\De_n\to0$ and $T_n\to\infty$. When $\la$ is known, we can even drop the assumption $T_n\to\infty$. The novelty is that these results hold without any kind of ergodicity or even recurrence assumption on the diffusion process.