Consistent inference for diffusions from low frequency measurements

Abstract

Let $({\mathit{X}}_{\mathit{t}})$ be a reflected diffusion process in a bounded convex domain in ${\mathbb{R}}^{\mathit{d}}$, solving the stochastic differential equation

$$\mathit{d}{\mathit{X}}_{\mathit{t}}=\nabla \mathit{f}({\mathit{X}}_{\mathit{t}})\mathit{d}\mathit{t}\mathbf{+}\sqrt{2\mathit{f}({\mathit{X}}_{\mathit{t}})}\mathit{d}{\mathit{W}}_{\mathit{t}},\mathit{t}\ge 0,$$

with ${\mathit{W}}_{\mathit{t}}$ a d-dimensional Brownian motion. The data ${\mathit{X}}_0,{\mathit{X}}_{\mathit{D}},\dots ,{\mathit{X}}_{\mathit{N}\mathit{D}}$ 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 ${\mathit{P}}_{\mathit{t},\mathit{f}}$. We prove injectivity theorems and stability inequalities for the maps $\mathit{f}\mapsto {\mathit{P}}_{\mathit{t},\mathit{f}}\mapsto {\mathit{P}}_{\mathit{D},\mathit{f}}$, $\mathit{t}<\mathit{D}$. 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.