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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; [email protected]
Dasha Loukianova
Affiliation:
Département de Mathématiques, Université d'Evry-Val d'Essonne, Bd François Mitterrand, 91025 Evry, France; [email protected]
Oleg Loukianov
Affiliation:
Département Informatique, IUT de Fontainebleau, Université Paris-Est, route Hurtault, 77300 Fontainebleau, France; [email protected]
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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].

Keywords

Diffusion processadaptive estimationregeneration methodmean square estimatormodel selectiondeviation inequalities
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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References

Banon, G., Nonparametric identification for diffusion processes. SIAM J. Control. Optim. 16 (1978) 380–395. CrossRef
Baraud, Y., Comte, F. and Viennet, G., Adaptive estimation in autoregression or β-mixing regression via model selection. Ann. Stat. 29 (2001) 839–875.
Y. Baraud, F. Comte and G. Viennet, Model selection for (auto-)regression with dependent data. ESAIM: P&S 5 (2001) 33–49. CrossRef
Barron, A., Birgé, L. and Massart, P., Risks bounds for model selection via penalization. Prob. Th. Rel. Fields 113 (1999) 301–413. CrossRef
Comte, F., Genon-Catalot, V. and Rozenholc, Y., Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli 13 (2007) 514–543. CrossRef
Dalalyan, A., Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Stat. 33 (2005) 507–2528. CrossRef
Dalalyan, A.S. and Kutoyants, Yu.A., Asymptotically efficient trend coefficient estimation for ergodic diffusion. Math. Meth. Stat. 11 (2002) 402–427.
S. Delattre, M. Hoffmann and M. Kessler, Dynamics adaptive estimation of a scalar diffusion. Prpublication PMA-762, Univ. Paris 6 (2002). Available at www.proba.jussieu.fr/mathdoc/preprints/. Mathematical Reviews (MathSciNet): MR1895888 Project Euclid: euclid.bj/1078866865.
Galtchouk, L. and Pergamenschikov, S., Sequential nonparametric adaptive estimation of the drift coefficient in the diffusion processes. Math. Meth. Stat. 10 (2001) 316–330.
Hoffmann, M., Adaptive estimation in diffusion processes. Stochastic Processes Appl. 79 (1999) 135–163. CrossRef
Yury A. Kutoyants, Statistical inference for ergodic diffusion processes. Springer Series in Statistics. London: Springer (2004).
Lepskii, O., One problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 (1999) 459–470.
G.G. Lorentz, M. von Golitschek and Y. Makovoz, Constructive approximation: advanced problems. Grundlehren der Mathematischen Wissenschaften 304. Berlin: Springer (1996).
Loukianova, D. and Loukianov, O., Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions. Ann. Inst. Henri Poincaré 44 (2008) 771–786. CrossRef
Löcherbach, E. and Loukianova, D., Nummelin, On splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stoch. Proc. Appl. 118 (2008) 301–1321. CrossRef
Löcherbach, E., Loukianova, D. and Loukianov, O., Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011) 425–449. CrossRef
T.D. Pham, Nonparametric estimation of the drift coefficient in the diffusion equation. Math. Operationsforsch. Statist., Ser. Statistics 1 (1981) 61–73.
B.L.S. Prakasa Rao, Statistical Inference for Diffusion Type Processes. London: Edward Arnold. MR1717690 (1999)
D. Revuz and M. Yor, Continuous martingales and Brownian motion. 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Berlin: Springer (2005).
Spokoiny, V.G., Adaptive drift estimation for nonparametric diffusion model. Ann. Stat. 28 (2000) 815–836. CrossRef
Veretennikov, A.Yu., On polynomial mixing bounds for stochastic differential equations. Stoch. Proc. Appl. 70 (1997) 115–127. CrossRef