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