On the long-term stability of bootstrap-type particle filters

In this paper we discuss optimal filtering in general state-space models (SSMs) and present novel theoretical results on the long-term stability of bootstrap-type particle filters. More specifically, we establish that the asymptotic variance of the particle estimates is uniformly bounded in time. On the contrary to most previous results of this type, which in general presuppose that the state space of the hidden state process is compact (an assumption that is rarely satisfied in practice), our very mild assumptions are satisfied for a large class of SSMs with possibly non-compact state space. In addition, we derive similar stability results for the Lp error of the particle estimates. Importantly, our results hold for misspecified models, i.e., we do not at all assume that the data entering into the particle filter originate from the model governing the dynamics of the particles or not even from an SSM.