Particle Filters for Partially Observed Diffusions

In this paper we introduce a novel particle filter scheme for a class of partiallyobserved multivariate diffusions. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the components of the multivariate diffusion and arrival times of a Poisson process whose intensity is a known function of the diffusion (Cox process). Unlike currently available methods, our particle filters do not require approximations of the transition and/or the observation density using time-discretisations. Instead, they build on recent methodology for the exact simulation of the diffusion process and the unbiased estimation of the transition density as described in Beskos et al. (2006b). We introduce the Generalised Poisson Estimator, which generalises the Poisson Estimator of Beskos et al. (2006b). A central limit theorem is given for our particle filter scheme.