Reversible jump MCMC for nonparametric drift estimation for diffusion processes

TL;DR

This paper introduces a reversible jump MCMC algorithm for nonparametric Bayesian estimation of diffusion process drift functions, capable of handling both continuous and discrete data observations, with demonstrated satisfactory results.

Contribution

It develops a novel reversible jump MCMC method for nonparametric drift estimation in diffusion processes, including model selection and data augmentation techniques.

Findings

Method provides accurate drift estimates in examples.

Algorithm effectively explores models of varying complexity.

Comparison shows competitive performance with existing methods.

Abstract

In the context of nonparametric Bayesian estimation a Markov chain Monte Carlo algorithm is devised and implemented to sample from the posterior distribution of the drift function of a continuously or discretely observed one-dimensional diffusion. The drift is modeled by a scaled linear combination of basis functions with a Gaussian prior on the coefficients. The scaling parameter is equipped with a partially conjugate prior. The number of basis function in the drift is equipped with a prior distribution as well. For continuous data, a reversible jump Markov chain algorithm enables the exploration of the posterior over models of varying dimension. Subsequently, it is explained how data-augmentation can be used to extend the algorithm to deal with diffusions observed discretely in time. Some examples illustrate that the method can give satisfactory results. In these examples a comparison is made with another existing method as well.