On pathwise convergence of particle & grid based nonlinear filters: Feller vs conditional regularity

We present a theoretical comparison of the state-of-the-art sufficient conditions required for pathwise (almost sure type of) convergence between grid based and particle approximate filters, as well as discuss the implications of these conditions on the specific mode of convergence achieved. Focusing on general Markov processes observed in conditionally Gaussian noise, we have recently shown that a sufficient condition for pathwise convergence of grid based filters is conditional regularity of stochastic kernels. The respective condition for almost sure convergence of particle filters is the well known Feller property. While our analysis shows that the comparison between the afore-mentioned conditions may be indeed inconclusive, we identify a large class of systems for which conditional regularity may hold true, whereas the Feller property cannot. This is achieved through a structural analysis of both sufficient conditions. This work can be summarized in that there provably exist system classes supported by either grid based or particle filtering approximations, but not necessarily by both; for systems supported by both, grid based filters exhibit a theoretical advantage in terms of convergence robustness.