Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60J60 Diffusion processes, See also {58G32}
- 62F12 Asymptotic properties of estimators
- 62M05 Markov processes: estimation

**Résumé:** We consider a diffusion process $X$ which is observed at times $i/n$
for $i=0,1,\ldots,n$, each observation being subject to a measurement
error. All errors are independent and centered Gaussian with known
variance $\r_n$. There is an unknown parameter within the diffusion
coefficient, to be estimated. In this first paper the
case when $X$ is indeed a Gaussian martingale is examined: we can prove
that the LAN property holds under quite weak smoothness assumptions,
with an explicit limiting Fisher information. What is perhaps the most
interesting is the rate at which this convergence takes place:
it is $1/\rn$ (as when there is no measurement error) when $\r_n$ goes fast
enough to $0$, namely $n\r_n$ is bounded. Otherwise, and provided the
sequence $\r_b$ itself is bounded, the rate is $(\r_n/n)^{1/4}$. In
particular if $\r_n=\r$ does not depend on $n$, we get a rate
$n^{-1/4}$.

**Mots Clés:** *Statistics of diffusions ; measurement errors ; LAN property*

**Date:** 2000-11-21

**Prépublication numéro:** *PMA-623*