Asymptotics of Ensemble Filters for Linear Stochastic Systems with Poisson-Sampled Observations

For continuous-time linear stochastic dynamical systems driven by Wiener processes, we consider the problem of designing ensemble filters when the observation process is randomly time-sampled. For the design of ensemble filters, we consider a class of continuous-discrete diffusion processes with additive Gaussian noise and several design parameters, which are used to describe the evolution of the individual particles in the ensemble. These particles are coupled through the empirical covariance, and in some cases empirical mean as well, and require less computations for implementation than the optimal ones based on solving Riccati differential equations. For different choices of parameters, we can recover some common design techniques from the literature. Our focus in this work is on analyzing the asymptotic (in time) performance of these filters for sufficiently large number of particles. Using appropriate analysis tools, we derive differential equations to describe the expectation of empirical mean and sample covariance of the ensemble filters with respect to the sampling process and noise. The solutions of these differential equations (describing empirical moments) are shown to converge asymptotically to the mean and covariance of the optimal filter under certain conditions on the mean sampling rate of the observation process, and as the number of particles tends to infinity.