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On estimating the diffusion coefficient from discrete observations

Part of: Markov processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Danielle Florens-Zmirou
Danielle Florens-Zmirou*
Affiliation:
Université de Paris Sud
*
∗ Postal address: CNRS Statistique Appliquée, URA D0743, Université Paris-Sud, Mathématiques Bât. 425, 91405 Orsay Cedex, France.
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Abstract

This paper is concerned with the problem of estimation for the diffusion coefficient of a diffusion process on R, in a non-parametric situation. The drift function can be unknown and considered as a nuisance parameter. We propose an estimator of σ based on discrete observation of the diffusion X throughout a given finite time interval. We describe the asymptotic behaviour of this estimator when the step of discretization tends to zero. We prove consistency and asymptotic normality, the rate of convergence to the normal law being a random variable linked to the local time of the diffusion or to its suitable discrete approximation. This can also be interpreted as a convergence to a mixture of normal law.

Keywords

DIFFUSIONSLOCAL TIMEVARIANCE ESTIMATIONDISCRETIZATION

MSC classification

Primary: 62M05: Markov processes
Secondary: 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 4 , December 1993 , pp. 790 - 804
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

The author is also affiliated to CEREMADE, Université de Paris-IX-Dauphine.

References

[1] Azais, J. M. (1989) Approximation des trajectoires et temps local des diffusions. Ann. Inst. H. Poincaré 25, 175–194.Google Scholar
[2] Bannon, G. and Nguyen, H. T. (1987) Recursive estimation in diffusion models. SIAM J. Control Optim. 19, 676–685.Google Scholar
[3] Dacunha-Castelle, D. and Florens-Zmirou, D. (1986) Estimation of the coefficient of a diffusion from discrete observations. Stochastics 19, 263–284.Google Scholar
[4] Donhal, G. (1981) On estimating the diffusion coefficient. J. Appl. Prob. 24, 105–114.Google Scholar
[5] Florens-Zmirou, D. (1989) Estimation of the diffusion coefficient from a discretized observation. C. R. Acad. Sci. Paris Ser. I 309, 195–200.Google Scholar
[6] Florens-Zmirou, D. (1991) Statistics on crossings of discretized distributions and local time. Stoch. Proc. Appl. 39, 139–151.Google Scholar
[7] Friedman, A. (1964) Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
[8] Jacod, J. (1983) Théorèmes limites pour des processus. Lecture Notes in Mathematics 1117, Springer-Verlag, Berlin.Google Scholar
[9] Knight, F. B. (1971) A Reduction of Continuous Square Integrable Martingales to Brownian Motion. Lecture Notes in Mathematics 190, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[10] Kukoyants, Y. (1984) Parameter Estimation for Stochastic Processes. Heldermann.Google Scholar
[11] Lanska, V. (1979) Minimum contrast estimation in diffusion processes. J. Appl. Prob. 16, 65–75.Google Scholar
[12] Mackean, H. P. (1969) Stochastic Integrals. Academic Press, New York.Google Scholar
[13] Pham Dinh, T. D. (1979) Non-parametric estimation of the drift coefficient in the diffusion equation. Sem. Statist. Univ. Grenoble, pp. 103–129.Google Scholar
[14] Revuz, D. and Yor, M. (1988) Continuous Martingale Calculus. Springer-Verlag, Berlin.Google Scholar
[15] Yor, M. (1978) Temps locaux. Astérisque 52–53, 17–35.Google Scholar