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

Frank Meulen (), Moritz Schauer () and Jan Waaij ()
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Frank Meulen: TU Delft
Moritz Schauer: Leiden University
Jan Waaij: Korteweg-de Vries Institute for Mathematics

Statistical Inference for Stochastic Processes, 2018, vol. 21, issue 3, No 7, 603-628

Abstract: 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 $$L_2$$ L 2 -norm that are optimal up to a log factor. Contraction rates in $$L_p$$ L p -norms with $$p\in (2,\infty ]$$ p ∈ ( 2 , ∞ ] are derived as well.

Date: 2018
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DOI: 10.1007/s11203-017-9163-7

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