Bayesian estimation of incompletely observed diffusions

We present a general framework for Bayesian estimation of incompletely observed multivariate diffusion processes. Observations are assumed to be discrete in time, noisy and incomplete. We assume the drift and diffusion coefficient depend on an unknown parameter. A data-augmentation algorithm for drawing from the posterior distribution is presented which is based on simulating diffusion bridges conditional on a noisy incomplete observation at an intermediate time. The dynamics of such filtered bridges are derived and it is shown how these can be simulated using a generalised version of the guided proposals introduced in Schauer et al. (2016).