Particle Filtering for Diffusions Avoiding Time-Discretisations

Abstract

In this short communication we present our recent work on the construction of novel particle filters for a class of partially-observed continuous-time dynamic models where the signal is given by a multivariate diffusion process; details are deferred to [1]. Our approach directly covers 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 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 diffusion process and the unbiased estimation of the transition density as described in the recent article [2]. In particular, we require the Generalised Poisson Estimator, which is developed in [1]. Thus, our filters avoid the systematic biases caused by time-discretisations and they have significant computational advantages over alternative continuous-time filters. These advantages are supported by a central limit theorem.