Bayesian estimation of discretely observed diffusion processes using Wiener chaos expansion

We employ a Bayesian inference technique for discretely observed diffusion processes that arise as solutions of stochastic differential equations. Our aim is to estimate the parameters of the stochastic differential equation. To achieve this, we frame the estimation procedure as a missing data problem. In this framework, the complete dataset includes the theoretically continuous-time path between observed points. We propose augmenting the dataset and using a Gibbs sampler to derive Bayesian estimators for the parameters in cases where the diffusion process is observed discretely. The Gibbs sampler is integrated with a diffusion bridge simulation technique based on the Wiener chaos expansion. The methodology and its implementation are demonstrated through examples and simulation studies. We also present an application to actual data.