# Nonparametric adaptive estimation for integrated diffusions.

Abstract : We consider here nonparametric estimation for integrated diffusion processes. Let $(V_t)$ be a stationary and $\beta$-mixing diffusion with unknown drift and diffusion coefficient. The integrated process $X_t= \int_0^{t} V_s ds$ is observed at discrete times with regular sampling interval $\Delta$. For both the drift function and the diffusion coefficient of the unobserved diffusion $(V_t)$, we propose nonparametric estimators based on a penalized least square approach. Estimators are chosen among a collection of functions belonging to a finite dimensional space selected by an automatic data-driven method. We derive non asymptotic risk bounds for the estimators. Interpreting these bounds through the asymptotic framework of high frequency data, we show that our estimators reach the minimax optimal rates of convergence. The algorithms of estimation are implemented for several examples of diffusion models that can be exactly simulated.
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Stochastic Processes and their Applications, Elsevier, 2009, 119 (3), pp.811-834. <10.1016/j.spa.2008.04.009>
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https://hal.archives-ouvertes.fr/hal-00110510
Contributeur : Fabienne Comte <>
Soumis le : lundi 30 octobre 2006 - 14:11:16
Dernière modification le : mardi 11 octobre 2016 - 12:01:01
Document(s) archivé(s) le : mardi 6 avril 2010 - 21:17:17

### Citation

Fabienne Comte, Valentine Genon-Catalot, Yves Rozenholc. Nonparametric adaptive estimation for integrated diffusions.. Stochastic Processes and their Applications, Elsevier, 2009, 119 (3), pp.811-834. <10.1016/j.spa.2008.04.009>. <hal-00110510>

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