Adaptive nonparametric drift estimation for diffusion processes using Faber–Schauder expansions
Frank van der Meulen (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Moritz Schauer (Universiteit Leiden)
Jan van Waaij (Universiteit van Amsterdam)
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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 (Formula presented.)-norm that are optimal up to a log factor. Contraction rates in (Formula presented.)-norms with (Formula presented.) are derived as well.
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