Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals

TL;DR

This paper introduces a novel Bayesian framework for estimating parameters of multi-dimensional diffusion processes from discrete observations, using guided proposals and a Metropolis-Hastings sampler to efficiently sample diffusion bridges without discretization.

Contribution

It generalizes existing methods by providing a non-discretization approach with guided proposals and a time change, improving sampling efficiency for multidimensional diffusions.

Findings

Effective sampling of diffusion bridges demonstrated

Reduced discretization error with proposed time change and scaling

Numerical examples show improved performance over existing methods

Abstract

Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods.