Transport Inspired Particle Filters with Poisson-Sampled Observations in Gaussian Setting

Motivated by the need for developing computationally efficient solutions to filtering problem with limited information, this article develops particle filtering algorithms for continuous-time stochastic processes with time-sampled observation process. The state process is modeled by a continuous-time linear stochastic differential equation driven by Wiener process, and the observation process is a linear mapping of the state with additive Gaussian noise. For practical reasons, we assume that the observations are time-sampled and the underlying sampling process is a Poisson counter. With the aim of developing particle filters for this system, we first propose a mean-field type process which is an observation-driven stochastic differential equation such that the conditional distribution of this process given the observations coincides with the optimal filtering distribution. This model is then used to simulate a collection of particles which are driven only by the sample mean and sample covariance, without simulating the differential equation for the covariance matrix. It is shown that the dynamics of the sample mean and the sample covariance coincide with the optimal ones. An academic example is included for illustration.