Non-parametric estimation of the diffusion coefficient from noisy data

We consider a diffusion process (Xt)t ≥ 0, with drift b(x) and diffusion coefficient σ(x). At discrete times tk = kδ for k from 1 to M, we observe noisy data of the sample path, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Y_{k\delta}=X_{k\delta}+\varepsilon_{k}}$$\end{document} . The random variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(\varepsilon_{k}\right)}$$\end{document} are i.i.d, centred and independent of (Xt). The process (Xt )t ≥ 0 is assumed to be strictly stationary, β-mixing and ergodic. In order to reduce the noise effect, we split data into groups of equal size p and build empirical means. The group size p is chosen such that Δ = pδ is small whereas Mδ is large. Then, the diffusion coefficient σ2 is estimated in a compact set A in a non-parametric way by a penalized least squares approach and the risk of the resulting adaptive estimator is bounded. We provide several examples of diffusions satisfying our assumptions and we carry out various simulations. Our simulation results illustrate the theoretical properties of our estimators.