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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.

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