Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient

We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations on its solution over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the diffusion coefficient under which the data have been generated. For a specific choice of the prior based on the inverse gamma distribution, this posterior contraction rate turns out to be optimal for estimation of a H\"older-continuous diffusion coefficient with smoothness parameter $0<\lambda\leq 1.$ Our approach is straightforward to implement and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.