Jan van Waaij, HU Berlin
Adaptive nonparametric Bayesian methods for SDEs
We consider continuous time observations (Xt_0\le t\le T of the SDE dXt = \Theta(Xt)dt + dWt: Our goal is estimating the unknown drift function \Theta. Due to their numerical advantages, Bayesian methods are often used for this. In this talk I discuss optimal rates of convergence for these methods. I will start with an introduction to Bayesian methods for SDEs and what sufficient conditions are for posterior convergence. I show that the sufficient conditions are satisfied for the Gaussian process prior, which leads to optimal convergence rates, provided the smoothness of \Theta matches the smootness of the Gaussian process. Adaptivity can be obtained by equipping the hyperparameter(s) of the Gaussian process prior with an additional prior. We discuss several choices for such hyperparameters. A new promising approach to obtain adaptivity is empirical Bayes. Here the optimal hyperparameters for the Gaussian process prior are first estimated from the data and then the prior with those hyperparameters plugged- in is used for the inference. This talk is based on van Waaij, 2018 and joint work with Frank van der Meulen (TU Delft), Moritz Schauer (Leiden University) and Harry van Zanten (University of Amsterdam).