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We consider a nonparametric Bayesian approach to estimate the diffusioncoefficient of a stochastic differential equation given discrete timeobservations over a fixed time interval. As a prior on the diffusioncoefficient, we employ a histogram-type prior with piecewise constantrealisations on bins forming a partition of the time interval. Specifically,these constants are realizations of independent inverse Gamma distributedrandoma variables. We justify our approach by deriving the rate at which thecorresponding posterior distribution asymptotically concentrates around thedata-generating diffusion coefficient. This posterior contraction rate turnsout to be optimal for estimation of a H\"older-continuous diffusion coefficientwith smoothness parameter $0<\lambda\leq 1.$ Our approach is straightforward toimplement, as the posterior distributions turn out to be inverse Gamma again,and leads to good practical results in a wide range of simulation examples.Finally, we apply our method on exchange rate data sets.

Nonparametric Bayesian estimation of a H\"older continuous diffusion coefficient