Data Augmentation for Diffusions

The problem of formal likelihood-based (either classical or Bayesian) inference for discretely observed multidimensional diffusions is particularly challenging. In principle, this involves data augmentation of the observation data to give representations of the entire diffusion trajectory. Most currently proposed methodology splits broadly into two classes: either through the discretization of idealized approaches for the continuous-time diffusion setup or through the use of standard finite-dimensional methodologies discretization of the diffusion model. The connections between these approaches have not been well studied. This article provides a unified framework that brings together these approaches, demonstrating connections, and in some cases surprising differences. As a result, we provide, for the first time, theoretical justification for the various methods of imputing missing data. The inference problems are particularly challenging for irreducible diffusions, and our framework is correspondingly more complex in that case. Therefore, we treat the reducible and irreducible cases differently within the article. Supplementary materials for the article are available online.