Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Adaptive wavelet estimation of the diffusion coefficient under additive error measurements

M. Hoffmann, A. Munk, and J. Schmidt-Hieber

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Abstract

We study nonparametric estimation of the diffusion coefficient from discrete data, when the observations are blurred by additional noise. Such issues have been developed over the last 10 years in several application fields and in particular in high frequency financial data modelling, however mainly from a parametric and semiparametric point of view. This paper addresses the nonparametric estimation of the path of the (possibly stochastic) diffusion coefficient in a relatively general setting.

By developing pre-averaging techniques combined with wavelet thresholding, we construct adaptive estimators that achieve a nearly optimal rate within a large scale of smoothness constraints of Besov type. Since the diffusion coefficient is usually genuinely random, we propose a new criterion to assess the quality of estimation; we retrieve the usual minimax theory when this approach is restricted to a deterministic diffusion coefficient. In particular, we take advantage of recent results of Reiß (Ann. Statist. 39 (2011) 772–802) of asymptotic equivalence between a Gaussian diffusion with additive noise and Gaussian white noise model, in order to prove a sharp lower bound.

Résumé

On étudie l’estimation non-paramétrique du coefficient de diffusion à partir d’observations discrètes, lorsque les observations sont bruitées par un bruit additionnel. De tels problèmes se sont développés au cours des dix dernières années dans plusieurs champs d’application, en particuler pour la modélisation des données haute fréquence en finance, cependant plutôt d’un point de vue paramétrique ou semi-paramétrique. Ce travail concerne l’estimation de la trajectoire (éventuellement stochastique) du coefficient de diffusion dans un cadre relativement général.

En développant des techniques de pré-moyennage combinées avec du seuillage des coefficients d’ondelettes, nous contruisons des estimateurs adaptatifs qui atteignent une vitesse quasi-optimale parmi une vaste échelle de contraintes de régularité de type Besov. Puisque le coefficient de diffusion est souvent intrinsèquement aléatoire, nous proposons un nouveau critère pour qualifier la qualité d’estimation ; nous retrouvons la théorie minimax usuelle lorsque cette approche est restreinte à un coefficient de diffusion déterministe. En particulier, on exploite les résultats récents de Reiß (Ann. Statist. 39 (2011) 772–802) de l’équivalence asymptotique entre une diffusion gaussienne avec un bruit additif et le bruit blanc gaussien.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 4 (2012), 1186-1216.

Dates
First available in Project Euclid: 16 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1353098445

Digital Object Identifier
doi:10.1214/11-AIHP472

Mathematical Reviews number (MathSciNet)
MR3052408

Zentralblatt MATH identifier
1282.62078

Subjects
Primary: 62G99: None of the above, but in this section 62M99: None of the above, but in this section 60G99: None of the above, but in this section

Keywords
Adaptive estimation Besov spaces Diffusion processes Nonparametric regression Wavelet estimation

Citation

Hoffmann, M.; Munk, A.; Schmidt-Hieber, J. Adaptive wavelet estimation of the diffusion coefficient under additive error measurements. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 4, 1186--1216. doi:10.1214/11-AIHP472. https://projecteuclid.org/euclid.aihp/1353098445


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