Continuous-discrete smoothing of diffusions

TL;DR

This paper introduces the Backward Filtering Forward Guiding (BFFG) algorithm, a novel MCMC method for exact smoothing of discretely observed diffusion processes with noisy transformations, also enabling parameter estimation.

Contribution

The paper presents a new MCMC algorithm, BFFG, for exact smoothing of diffusions observed discretely with noise, extending previous guided proposals.

Findings

Efficient smoothing in challenging diffusion problems

Extension to include parameter estimation

Demonstrated effectiveness on complex examples

Abstract

Suppose X is a multivariate diffusion process that is observed discretely in time. At each observation time, a transformation of the state of the process is observed with noise. The smoothing problem consists of recovering the path of the process, consistent with the observations. We derive a novel Markov Chain Monte Carlo algorithm to sample from the exact smoothing distribution. The resulting algorithm is called the Backward Filtering Forward Guiding (BFFG) algorithm. We extend the algorithm to include parameter estimation. The proposed method relies on guided proposals introduced in Schauer et al. (2017). We illustrate its efficiency in a number of challenging problems.