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May 2007 Penalized nonparametric mean square estimation of the coefficients of diffusion processes
Fabienne Comte, Valentine Genon-Catalot, Yves Rozenholc
Bernoulli 13(2): 514-543 (May 2007). DOI: 10.3150/07-BEJ5173


We consider a one-dimensional diffusion process $(X_t)$ which is observed at $n+1$ discrete times with regular sampling interval $Δ$. Assuming that $(X_t)$ is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms.


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Fabienne Comte. Valentine Genon-Catalot. Yves Rozenholc. "Penalized nonparametric mean square estimation of the coefficients of diffusion processes." Bernoulli 13 (2) 514 - 543, May 2007.


Published: May 2007
First available in Project Euclid: 18 May 2007

zbMATH: 1127.62067
MathSciNet: MR2331262
Digital Object Identifier: 10.3150/07-BEJ5173

Keywords: adaptive estimation , Diffusion processes , discrete time observations , Drift and diffusion coefficients , mean square estimator , Model selection , Penalized contrast , retrospective simulation

Rights: Copyright © 2007 Bernoulli Society for Mathematical Statistics and Probability

Vol.13 • No. 2 • May 2007
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