  We consider the problem of making nonparametric inference in
multi-dimensional diffusion models from low-frequency data. Statistical
analysis in this setting is notoriously challenging due to the intractability
of the likelihood and its gradient, and computational methods have thus far
largely resorted to expensive simulation-based techniques. In this article, we
propose a new computational approach which is motivated by PDE theory and is
built around the characterisation of the transition densities as solutions of
the associated heat (Fokker-Planck) equation. Employing optimal regularity
results from the theory of parabolic PDEs, we prove a novel characterisation
for the gradient of the likelihood. Using these developments, for the nonlinear
inverse problem of recovering the diffusivity (in divergence form models), we
then show that the numerical evaluation of the likelihood and its gradient can
be reduced to standard elliptic eigenvalue problems, solvable by powerful
finite element methods. This enables the efficient implementation of a large
class of statistical algorithms, including (i) preconditioned Crank-Nicolson
and Langevin-type methods for posterior sampling, and (ii) gradient-based
descent optimisation schemes to compute maximum likelihood and
maximum-a-posteriori estimates. We showcase the effectiveness of these methods
via extensive simulation studies in a nonparametric Bayesian model with
Gaussian process priors. Interestingly, the optimisation schemes provided
satisfactory numerical recovery while exhibiting rapid convergence towards
stationary points despite the problem nonlinearity; thus our approach may lead
to significant computational speed-ups. The reproducible code is available
online at https://github.com/MattGiord/LF-Diffusion.
