Estimation of hyperbolic diffusion using the Markov chain Monte Carlo method

Abstract

this paper we propose a Bayesian method to estimate the hyperbolic diffusion model. The approach is based on the Markov chain Monte Carlo (MCMC) method with the likelihood of the discretized process as the approximate posterior likelihood. We demonstrate that the MCMC method provides a useful tool in analysing hyperbolic diffusions. In particular, quantities of posterior distributions obtained from the MCMC outputs can be used for statistical inference. The MCMC method based on the Milstein scheme is found to perform well with good mixing properties, while the Euler scheme is unsatisfactory. Our simulation study shows that the hyperbolic diffusion exhibits many of the stylized facts about asset returns documented in the discrete-time financial econometrics literature, such as the Taylor effect, a slowly declining autocorrelation function of the squared returns, and thick tails.