Exponential forgetting and geometric ergodicity for optimal filtering in general state-space models

State-space models are a very general class of time series capable of modeling-dependent observations in a natural and interpretable way. We consider here the case where the latent process is modeled by a Markov chain taking its values in a continuous space and the observation at each point admits a distribution dependent of both the current state of the Markov chain and the past observation. In this context, under given regularity assumptions, we establish that (1) the filter, and its derivatives with respect to some parameters in the model, have exponential forgetting properties and (2) the extended Markov chain, whose components are the latent process, the observation sequence, the filter and its derivatives is geometrically ergodic. The regularity assumptions are typically satisfied when the latent process takes values in a compact space.