Markov chain Monte Carlo for exact inference for diffusions

TL;DR

This paper introduces exact Markov chain Monte Carlo methods for inference in discretely-sampled diffusions, eliminating discretisation errors and improving performance over existing high-frequency imputation techniques.

Contribution

It develops novel exact MCMC algorithms applicable to a broad class of diffusions, incorporating reparametrisations and auxiliary sampling to enhance efficiency.

Findings

Methods are theoretically sound and empirically effective.

Compared favorably to high-frequency imputation in experiments.

Uncovered connections between different inference approaches.

Abstract

We develop exact Markov chain Monte Carlo methods for discretely-sampled, directly and indirectly observed diffusions. The qualification "exact" refers to the fact that the invariant and limiting distribution of the Markov chains is the posterior distribution of the parameters free of any discretisation error. The class of processes to which our methods directly apply are those which can be simulated using the most general to date exact simulation algorithm. The article introduces various methods to boost the performance of the basic scheme, including reparametrisations and auxiliary Poisson sampling. We contrast both theoretically and empirically how this new approach compares to irreducible high frequency imputation, which is the state-of-the-art alternative for the class of processes we consider, and we uncover intriguing connections. All methods discussed in the article are tested on typical examples.