Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process

¹ 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
² Département de Mathématiques, Université d'Evry-Val d'Essonne, Bd François Mitterrand, 91025 Evry, France; dasha.loukianova@univ-evry.fr
³ Département Informatique, IUT de Fontainebleau, Université Paris-Est, route Hurtault, 77300 Fontainebleau, France; oleg@iut-fbleau.fr

Received: 25 March 2009
Revised: 6 October 2009

Abstract

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].