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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 process, adaptive estimation, regeneration method, mean square estimator, model selection, deviation inequalities
@article{PS_2011__15__197_0,
author = {L\"ocherbach, Eva and Loukianova, Dasha and Loukianov, Oleg},
title = {Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process},
journal = {ESAIM: Probability and Statistics},
pages = {197--216},
publisher = {EDP-Sciences},
volume = {15},
year = {2011},
doi = {10.1051/ps/2009016},
mrnumber = {2870512},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2009016/}
}
TY - JOUR AU - Löcherbach, Eva AU - Loukianova, Dasha AU - Loukianov, Oleg TI - Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process JO - ESAIM: Probability and Statistics PY - 2011 SP - 197 EP - 216 VL - 15 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps/2009016/ DO - 10.1051/ps/2009016 LA - en ID - PS_2011__15__197_0 ER -
%0 Journal Article %A Löcherbach, Eva %A Loukianova, Dasha %A Loukianov, Oleg %T Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process %J ESAIM: Probability and Statistics %D 2011 %P 197-216 %V 15 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps/2009016/ %R 10.1051/ps/2009016 %G en %F PS_2011__15__197_0
Löcherbach, Eva; Loukianova, Dasha; Loukianov, Oleg. Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 197-216. doi: 10.1051/ps/2009016
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