In this article we summarise recent work in modelling and estimation of continuous-time jump models for application in tracking scenarios. The models are constructed such that random jumps occur in the driving function (typically the applied force on an object being tracked) at random times, and in general asynchronously with the observation times. The sojourn times between jumps are modelled as general distributions (here gamma or shifted gamma), and hence we are in the class of semi-Markov models, since the arrival times do not form a Poisson point process (see Li and Jilkov, [IEEE Trans. Aerospace and Electronic Systems, 39(4), 2003] for an overview of such models in the tracking setting). In contrast with other such models in the tracking literature we allow a fully continuous random set of manoeuvre parameters, rather than a discrete set of switching models, and deterministic paths during the sojourns, obeying a set of nonlinear kinematic equations for point mass motion, thus modelling the path of the object in a smooth and parsimonious fashion. These models are aimed at capturing the highly random manoeuvres of real objects in a simple way. Estimation is carried out using a Variable Rate Particle Filter (VRPF) that parameterises the model explicitly in terms of the jump times and their parameters [Godsill and Vermaak, Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., 2004, and Proc. IEEE Stat. Sig. Proc., Bordeaux, July 2005.] Extensions to the models and algorithms are also presented that allow for a diffusion component of the model, which captures continuous random disturbances to the object in addition to jumps. These are illustrated in a 3-dimensional linear Gaussian setting where the entire path, except for jump times, may be marginalised, hence making a more efficient and effective particle filter.
