Abstract
We study the performance of nonparametric Bayes procedures for one-dimensional diffusions with periodic drift. We improve existing convergence rate results for Gaussian process (GP) priors with fixed hyper parameters. Moreover, we exhibit several possibilities to achieve adaptation to smoothness. We achieve this by considering hierarchical procedures that involve either a prior on a multiplicative scaling parameter, or a prior on the regularity parameter of the GP.
Highlights
Various papers have recently considered nonparametric Bayes procedures for one-dimensional stochastic differential equations (SDEs) with periodic drift. This is motivated among others by problems in which SDEs are used for the dynamic modelling of angles in different contexts
A first option we explore is putting a prior on the multiplicative constant η in (1.2), instead of taking it fixed as in Papaspiliopoulos et al (2012) and Pokern et al (2013)
Our first main result deals with the case that the scaling parameter and the regularity parameter of the Gaussian process (GP) are fixed, positive constants
Summary
Various papers have recently considered nonparametric Bayes procedures for one-dimensional stochastic differential equations (SDEs) with periodic drift. In the present paper we follow instead the approach of van der Meulen et al (2006), which is essentially an adaptation to the SDE case of the general “testing approach” which has become well known in Bayesian nonparametrics These ideas, combined with results about the asymptotic behaviour of the so-called periodic diffusion local time from Pokern et al (2013), allow us to obtain the new, sharp result for the GP prior with precision (1.2). The paper van der Meulen et al (2014) considers a related but different computational strategy, which combines a prior on the multiplicative constant with a random truncation of the series that defines the Gaussian prior.
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