Adaptive nonparametric drift estimation for diffusion processes using Faber\u2013Schauder expansions

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 et al. (Comput Stat Data Anal 71:615–632, 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 this 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. Contraction rates in L_p-norms with pin (2,infty ] are derived as well.