Diffusions with measurement errors. II. Optimal estimators
ESAIM: Probability and Statistics, Tome 5 (2001), pp. 243-260.

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 ρ 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.

Classification : 60J60,  62F12,  62M05
Mots clés : statistics of diffusions, measurement errors, LAN property
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     title = {Diffusions with measurement errors. {II.} {Optimal} estimators},
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Gloter, Arnaud; Jacod, Jean. Diffusions with measurement errors. II. Optimal estimators. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 243-260. http://www.numdam.org/item/PS_2001__5__243_0/

[1] G. Dohnal, On estimating the diffusion coefficient. J. Appl. Probab. 24 (1987) 105-114. | MR 876173 | Zbl 0615.62109

[2] V. Genon-Catalot and J. Jacod, On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 119-153. | Numdam | Zbl 0770.62070

[3] A. Gloter and J. Jacod, Diffusion with measurement error. I. Local Asymptotic Normality (2000). | Numdam | MR 1875672 | Zbl 1008.60089

[4] J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin (1987). | MR 959133 | Zbl 0635.60021

[5] J. Jacod, On continuous conditional Gaussian martingales and stable convergence in law, Séminaire Proba. XXXI. Springer-Verlag, Berlin, Lecture Notes in Math. 1655 (1997) 232-246. | Numdam | MR 1478732 | Zbl 0884.60038

[6] M.B. Malyutov and O. Bayborodin, Fitting diffusion and trend in noise via Mercer expansion, in Proc. 7th Int. Conf. on Analytical and Stochastic Modeling Techniques. Hamburg (2000).

[7] A. Renyi, On stable sequences of events. Sankyā Ser. A 25 (1963) 293-302. | MR 170385 | Zbl 0141.16401