A Simple Bayesian Method for the Analysis of Diffusion Processes

This paper introduces a new Bayesian method for the analysis of discretely sampled diffusion processes. The method, which is termed high frequency augmentation (HFA), is a simple numerical method that is applicable to a wide variety of univariate or multivariate diffusion processes. It is furthermore useful when observations are irregularly observed, when one or more elements of the multivariate process are latent, or when microstructure effects add error to the observed data. The Markov chain-Monte Carlo-based procedure can be used to attain the posterior distributions of the parameters of the drift and diffusion functions as well as the posteriors of missing or latent data. Several examples are explored. First, the posterior of the parameters of a geometric Brownian motion is attained using HFA and compared with that obtained using the analytical methods available for this relatively simple process. Second, a stochastic volatility model is estimated on a sample of S&P 500 returns, a problem for which posteriors are analytically intractable. Lastly, it is shown how the method can be extended to estimate an interest rate process using data that suffer from severe rounding.