Diffusions with measurement errors. I. Local asymptotic normality

Gloter, Arnaud; Jacod, Jean

ESAIM: Probability and Statistics, Tome 5 (2001), pp. 225-242.

We consider a diffusion process $X$ which is observed at times $i/n$ for $i=0,1,...,n$, each observation being subject to a measurement error. All errors are independent and centered gaussian with known variance ${\rho }_n$. 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 $1/\sqrt{n}$ (as when there is no measurement error) when ${\rho }_n$ goes fast enough to $0$, namely $n{\rho }_n$ is bounded. Otherwise, and provided the sequence ${\rho }_n$ itself is bounded, the rate is ${({\rho }_n/n)}^{1/4}$. In particular if ${\rho }_n=\rho$ does not depend on $n$, we get a rate $n^{-1/4}$.

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