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Penalized nonparametric drift estimationfor a continuously observed one-dimensional diffusion process

Published online by Cambridge University Press:  05 January 2012

Eva Löcherbach ,
Dasha Loukianova  and
Oleg Loukianov
Eva Löcherbach
Affiliation:
Centre de Mathématiques, Faculté de Sciences et Technologie, Université Paris-Est Val-de-Marne, 61 avenue du Général de Gaulle, 94010 Créteil, France; locherbach@univ-paris12.fr
Dasha Loukianova
Affiliation:
Département de Mathématiques, Université d'Evry-Val d'Essonne, Bd François Mitterrand, 91025 Evry, France; dasha.loukianova@univ-evry.fr
Oleg Loukianov
Affiliation:
Département Informatique, IUT de Fontainebleau, Université Paris-Est, route Hurtault, 77300 Fontainebleau, France; oleg@iut-fbleau.fr

Abstract

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Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in stationary regime neither to be exponentially β-mixing. This is possible thanks to the use of a new polynomial inequality in the ergodic theorem [E. Löcherbach, D. Loukianova and O. Loukianov, Ann. Inst. H. Poincaré Probab. Statist. 47 (2011) 425–449].

Keywords

Diffusion processadaptive estimationregeneration methodmean square estimatormodel selectiondeviation inequalities

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 197 - 216
Copyright
© EDP Sciences, SMAI, 2011

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