Diffusions with measurement errors. I. Local Asymptotic Normality

¹ G.R.A.P.E., UMR 5113 du CNRS, Université Montesquieu (Bordeaux), Avenue Léon Duguit, 33608 Pessac, France; gloter@montesquieu.u-bordeaux.fr.
² Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 du CNRS, Université Paris 6, 4 place Jussieu, 75252 Paris, France; jj@ccr.jussieu.fr.

Received: 16 February 2001
Revised: 24 October 2001

Abstract

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 p_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 (as when there is no measurement error) when p_n goes fast enough to 0, namely np_n is bounded. Otherwise, and provided the sequence p_n itself is bounded, the rate is (p_n / n)^1/4. In particular if p_n = p does not depend on n, we get a rate n^-1/4.