Particle Filtering for a Class of State-Space Models with Low and Degenerate Observational Noise

We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. We focus on the case where the observation equation is a linear function of the state and that additive noise is low or degenerate, however, we place minimal assumptions on the hidden state process. In this scenario we derive new particle filtering (PF) algorithms and, under assumptions, in such a way that as the noise becomes more degenerate a PF which approximates the low noise filtering problem provably inherits the properties of the PF used in the degenerate case. We extend our framework to the case where the hidden states are drawn from a diffusion process. In this scenario we develop new PFs which are robust to both low noise and fine levels of time discretization. We illustrate our algorithms numerically on several examples.