Adaptive nonparametric drift estimation for diffusion processes using Faber-Schauder expansions

TL;DR

This paper develops an adaptive nonparametric Bayesian method for estimating the drift of diffusion processes using Faber-Schauder expansions, achieving near-optimal convergence rates for smooth true drifts.

Contribution

It introduces a prior based on truncated and scaled Faber-Schauder series with Gaussian coefficients and analyzes its frequentist asymptotic properties.

Findings

Posterior contraction rates are optimal up to a log factor in L2-norm.

Contraction rates are also derived for Lp-norms with p in (2,∞].

The method adapts to the smoothness of the true drift.

Abstract

We consider the problem of nonparametric estimation of the drift of a continuously observed one-dimensional diffusion with periodic drift. Motivated by computational considerations, van der Meulen e.a. (2014) defined a prior on the drift as a randomly truncated and randomly scaled Faber-Schauder series expansion with Gaussian coefficients. We study the behaviour of the posterior obtained from the prior from a frequentist asymptotic point of view. If the true data generating drift is smooth, it is proved that the posterior is adaptive with posterior contraction rates for the $L_2$-norm that are optimal up to a log factor. Moreover, contraction rates in $L_p$-norms with $p\in (2,\infty]$ are derived as well.