Diffusions with measurement errors. I. Local Asymptotic Normality

Free access article

Issue		ESAIM: PS Volume 5, 2001


Page(s)		225 - 242
DOI		10.1051/ps:2001110

DOI: 10.1051/ps:2001110

ESAIM: P&S, December 2001, Vol. 5, pp. 225-242

Diffusions with measurement errors. I. Local Asymptotic Normality

Arnaud Gloter¹ and Jean Jacod²

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 February 16, 2001. Revised October 24, 2001)

Abstract
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 $\rho_n$ . 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.

AMS Subject: 60J60, 62F12, 62M05.

Key words: Statistics of diffusions, measurement errors, LAN property.

© EDP Sciences, SMAI 2001

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