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.

References in corpus (5)

• Rates of contraction of posterior distributions based on Gaussian process priors
• Convergence rates of posterior distributions for noniid observations
• Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$
• Sharp adaptive estimation of the drift function for ergodic diffusions
• Gaussian process methods for one-dimensional diffusions: optimal rates and adaptation

Cited by in corpus (1)

• A piecewise deterministic Monte Carlo method for diffusion bridges