Inference for diffusions from low frequency measurements

Let $(X_t)$ be a reflected diffusion process in a bounded convex domain in $\mathbb R^d$, solving the stochastic differential equation $$dX_t = \nabla f(X_t) dt + \sqrt{2f (X_t)} dW_t, ~t \ge 0,$$ with $W_t$ a $d$-dimensional Brownian motion. The data $X_0, X_D, \dots, X_{ND}$ 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 solve the non-linear inverse problem of inferring the diffusivity $f$ and the associated transition operator $P_{t,f}$. We prove injectivity theorems and stability estimates for the maps $f \mapsto P_{t,f} \mapsto P_{D,f}, t<D$. Using these estimates we then 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 `fast convergence' and the `hot spots' conjecture from spectral geometry.