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Schémas de discrétisation anticipatifs et estimation du paramètre de dérive d'une diffusion

Published online by Cambridge University Press:  15 August 2002

Sandie Souchet Samos
Sandie Souchet Samos*
Affiliation:
Université Paris 1, Centre Pierre Mendès France, 90 rue de Tolbiac, 75013 Paris, France; e-mail: ssouchet@caramail.com
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Abstract

Let YT = (Yt)t∈[0,T] be a real ergodic diffusion process which drift depends on an unkown parameter $\theta_{0}\in \mathbb{R}^{p}$. Our aim is to estimate θ0 from a discrete observation of the process YT, (Ykδ)k=0,n, for a fixed and small δ, as T = nδ goes to infinity. For that purpose, we adapt the Generalized Method of Moments (see Hansen) to the anticipative and approximate discrete-time trapezoidal scheme, and then to Simpson's. Under some general assumptions, the trapezoidal scheme (respectively Simpson's scheme) provides an estimation of θ0 with a bias of order δ2 (resp. δ4). Moreover, this estimator is asymptotically normal. These results generalize Bergstrom's [1], which were obtained for a Gaussian diffusion process, which drift is linear in θ.

Keywords

Schéma du trapèzeschéma de Simpsonschéma anticipatifdiffusion ergodiqueestimation par variables instrumentalesméthode des moments généraliséscontrastebiais d'estimationefficacité asymptotique en variance.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 4 , 2000 , pp. 233 - 258
Copyright
© EDP Sciences, SMAI, 2000

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References

A.R. Bergstrom, Statistical inference in Continuous Time Series, in Statistical inference in Continuous Time Economic Models, Bergstrom, Ed., North Holland, Amsterdam (1976).
Bibby, B.M. et Sorensen, M., Martingale Estimation Functions for Discretely Observed Diffusion Processes. Bernoulli 1 (1995) 17-39. CrossRef
D. Dacunha-Castelle et M. Duflo, Probabilité et Statistiques. Tome 2, 2e Ed. Masson (1993).
Dacunha-Castelle, D. et Florens-Zmirou, D., Estimation of the coefficient of a diffusion from discrete observations. Stochastics 19 (1986) 263-284. CrossRef
Florens-Zmirou, D., Approximate discrete schemes for statistics of diffusion processes. Statistics 20 (1989) 547-557. CrossRef
C. Gourieroux et A. Monfort, Statistique et Modèles Économétriques. Tome 1. Economica.
Hansen, L., Large Sample Properies of Generalized Method of Moments Estimators. Econometrica 50 (1982) 1029-1054. CrossRef
Hansen, L. et Singleton, K., Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models. Econometrica 50 (1982) 1269-1286. CrossRef
I. Karatzas et S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd Ed. Springer (1996).
Kessler, M., Estimation of an ergodic diffusion from discrete observations. Scand. J. Stat. 24 (1997) 211-229. CrossRef
M. Kessler, Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process. Research Reports 336, Department of theoretical statistics, University of Aarhus (1995).
M. Kessler et M. Sorensen, Estimating Equations Based on Eigenfunctions for a Discretely Observed Diffusion Process. Research Reports 332, Department of theoretical statistics, University of Aarhus (1995).
P.E. Kloeden et E. Platen, Numerical Solution of Stochastic Differential Equations. Springer (1995).
Yu.A. Kutoyants, Parameter estimation for stochastic processes. Heldermann Verlag, Berlin, Research and Exposition in Math. 6 (1984).
R.S. Liptser et A.N. Shiryaev, Statistics of random processes. Tomes 1, 2. Springer-Verlag (1977).
W.H. Press, S.A. Teukolskey, W.T. Vetterling et B.P. Flannery, Numerical Recipes in C, 2nd Ed. Cambridge University Press, 132-133.
Prakasa-Rao, B.L.S., Asymptotic theory for non linear least squares estimator for diffusion proceses. Math. Operationsforsch. Statist Ser. Berlin 14 (1983) 195-209.
Pedersen, A.R., Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes. Bernoulli 1 (1995) 257-279. CrossRef
Pedersen, A.R., A new approch to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist. 22 (1995) 55-71.
J.D. Sargan, Some discrete approximations to continuous times stochastics models, in Statistical inference in Continuous Time Economic Models. Bergstrom, Ed., North Holand, Amsterdam (1976) 27-80.
M. Sorensen, Estimating functions for discretely observed diffusions: A review. Research Reports 348, Department of theoretical statistics, University of Aarhus (1996).
Yoshida, N., Estimation for diffusion processes from discrete observations. J. Multivariate Anal. 41 (1992) 220-242. CrossRef