Title: Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient,
Abstract: We consider a nonparametric Bayesian approach to estimate the diffusion
coefficient of a stochastic differential equation given discrete time
observations 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. Specifically,
these constants are realizations of independent inverse Gamma distributed
randoma variables. We justify our approach by deriving the rate at which the
corresponding posterior distribution asymptotically concentrates around the
data-generating diffusion coefficient. This posterior contraction rate turns
out to be optimal for estimation of a Hölder-continuous diffusion coefficient
with smoothness parameter $0<\lambda\leq 1.$ Our approach is straightforward to
implement, 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.