Adaptive wavelet estimation of the diffusion coefficient under additive error measurements

TL;DR

This paper introduces an adaptive wavelet-based method for nonparametric estimation of diffusion coefficients from noisy discrete data, achieving near-optimal rates across various smoothness levels in a general setting.

Contribution

It develops a novel combination of pre-averaging and wavelet thresholding for adaptive estimation of stochastic diffusion coefficients under additive noise.

Findings

Achieves nearly optimal convergence rates for a wide class of smoothness.

Proposes a new criterion for assessing the quality of stochastic diffusion coefficient estimates.

Establishes sharp lower bounds using asymptotic equivalence with Gaussian white noise models.

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 [33] of asymptotic equivalence between a Gaussian diffusion with additive noise and Gaussian white noise model, in order to prove a sharp lower bound.