Consistent inference for diffusions from low frequency measurements

Let $(Xt)$ be a reflected diffusion process in a bounded convex domain in $\mathbb R^d$, solving the stochastic differential equation $$dXt = \nabla f(Xt) dt + \sqrt{2f (Xt)} dWt, ~t \ge 0,$$ with $Wt$ a $d$-dimensional Brownian motion. The data $X0, XD, \dots, X{ND}$ 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 $P{t,f}$. We prove injectivity theorems and stability inequalities for the maps $f \mapsto P{t,f} \mapsto P{D,f}, t<D$. 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.