Consistent inference for diffusions from low frequency measurements

TL;DR

This paper develops methods to reliably infer the diffusivity function of a reflected diffusion process from low-frequency discrete measurements, establishing theoretical guarantees and Bayesian algorithms with optimal convergence rates.

Contribution

It proves injectivity and stability results for the inverse problem, and introduces Bayesian algorithms with Gaussian process priors that are statistically consistent and rate-optimal.

Findings

Injectivity and stability theorems for the inverse map from transition operators to diffusivity.

Bayesian algorithms with Gaussian process priors are statistically consistent.

Established optimal convergence rates for the proposed inference methods.

Abstract

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 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.