Consistent inference for diffusions from low frequency measurements

Let (Xt) be a reflected diffusion process in a bounded convex domain in Rd, solving the stochastic differential equation dXt=∇f(Xt)dt+ 2f(Xt)dWt,t≥0, with Wt a d-dimensional Brownian motion. The data X0,XD,…,XND consist of discrete measurements and the time interval D between consecutive observations is fixed so that one cannot 'zoom' into the observed path of the process. The goal is to infer the diffusivity f and the associated transition operator Pt,f. We prove injectivity theorems and stability inequalities for the maps f↦Pt,f↦PD,f, t<D. Using these estimates, we establish the statistical consistency of a class of Bayesian algorithms based on Gaussian process priors for the infinite-dimensional parameter f, and show optimality of some of the convergence rates obtained. We discuss an underlying relationship between the degree of ill-posedness of this inverse problem and the 'hot spots' conjecture from spectral geometry.