Adaptive wavelet estimation of the diffusion coefficient under additive error measurements
Johannes, Schmidt-Hieber; Munk, Axel; Hoffmann, Marc (2012), Adaptive wavelet estimation of the diffusion coefficient under additive error measurements, Annales de l'I.H.P. Probabilités et Statistiques, 48, 4, p. 1186-1216. http://dx.doi.org/10.1214/11-AIHP472
Type
Article accepté pour publication ou publiéExternal document link
http://hal.archives-ouvertes.fr/hal-00779763Date
2012Journal name
Annales de l'I.H.P. Probabilités et StatistiquesVolume
48Number
4Publisher
Institut Henri Poincaré
Pages
1186-1216
Publication identifier
Metadata
Show full item recordAbstract (EN)
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.Subjects / Keywords
Wavelet estimation; Nonparametric regression; Diffusion processes; Besov spaces; Adaptive estimationRelated items
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