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

Eva Löcherbach; Dasha Loukianova; Oleg Loukianov

ESAIM: Probability and Statistics (2012)

- Volume: 15, page 197-216
- ISSN: 1292-8100

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topLöcherbach, Eva, Loukianova, Dasha, and Loukianov, Oleg. "Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process." ESAIM: Probability and Statistics 15 (2012): 197-216. <http://eudml.org/doc/222472>.

@article{Löcherbach2012,

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].},

author = {Löcherbach, Eva, Loukianova, Dasha, Loukianov, Oleg},

journal = {ESAIM: Probability and Statistics},

keywords = {Diffusion process; adaptive estimation; regeneration method; mean square estimator; model selection; deviation inequalities; diffusion process},

language = {eng},

month = {1},

pages = {197-216},

publisher = {EDP Sciences},

title = {Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process},

url = {http://eudml.org/doc/222472},

volume = {15},

year = {2012},

}

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

DA - 2012/1//

PB - EDP Sciences

VL - 15

SP - 197

EP - 216

AB -
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].

LA - eng

KW - Diffusion process; adaptive estimation; regeneration method; mean square estimator; model selection; deviation inequalities; diffusion process

UR - http://eudml.org/doc/222472

ER -

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