Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class.
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The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of convergence of the Nadaraya–Watson estimator in the case of a locally Hölder-continuous drift.
@article{1217964119,
author = {Loukianova, D. and Loukianov, O.},
title = {Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {44},
number = {2},
year = {2008},
pages = { 771-786},
language = {en},
url = {http://dml.mathdoc.fr/item/1217964119}
}
Loukianova, D.; Loukianov, O. Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions. Ann. Inst. H. Poincaré Probab. Statist., Tome 44 (2008) no. 2, pp. 771-786. http://gdmltest.u-ga.fr/item/1217964119/