Research Article

Diffusions with measurement errors. II. Optimal estimators

Gloter, Arnaud^a1 and Jacod, Jean^a2

^a1 G.R.A.P.E., UMR 5113 du CNRS, Université Montesquieu (Bordeaux), Avenue Léon Duguit, 33608 Pessac, France; gloter@montesquieu.u-bordeaux.fr.

^a2 Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 du CNRS, Université Paris 6, 4 place Jussieu, 75252 Paris, France; jj@ccr.jussieu.fr.

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 to estimate within the diffusion coefficient. In this second paper we construct estimators which are asymptotically optimal when the process X is a Gaussian martingale, and we conjecture that they are also optimal in the general case.

(Received February 2 2001)

(Revised October 24 2001)

(Online publication August 15 2002)

Key Words:

• Statistics of diffusions;
• measurement errors;
• LAN property.

Mathematics Subject Classification:

• 60J60;
• 62F12;
• 62M05