Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

Abstract

We consider nonparametric Bayesian inference in a reflected diffusionmodel dXt = b(Xt)dt + σ(Xt)dWt , with discretely sampled observationsX0,X, . . . , Xn. We analyse the nonlinear inverse problem correspondingto the “low frequency sampling” regime where >0 is fixed and n→∞.A general theorem is proved that gives conditions for prior distributions on the diffusion coefficient σ and the drift function b that ensure minimaxoptimal contraction rates of the posterior distribution over Hölder–Sobolevsmoothness classes. These conditions are verified for natural examples ofnonparametric random wavelet series priors. For the proofs, we derive newconcentration inequalities for empirical processes arising from discretely observeddiffusions that are of independent interest.