A Multiresolution Method for Parameter Estimation of Diffusion Processes

Diffusion process models are widely used in science, engineering, and finance. Most diffusion processes are described by stochastic differential equations in continuous time. In practice, however, data are typically observed only at discrete time points. Except for a few very special cases, no analytic form exists for the likelihood of such discretely observed data. For this reason, parametric inference is often achieved by using discrete-time approximations, with accuracy controlled through the introduction of missing data. We present a new multiresolution Bayesian framework to address the inference difficulty. The methodology relies on the use of multiple approximations and extrapolation and is significantly faster and more accurate than known strategies based on Gibbs sampling. We apply the multiresolution approach to three data-driven inference problems, one of which features a multivariate diffusion model with an entirely unobserved component.