Exponential forgetting and geometric ergodicity in hidden Markov models

We consider a hidden Markov model with multidimensional observations, and with misspecification, i.e., the assumed coefficients (transition probability matrix and observation conditional densities) are possibly different from the I,lte coefficients. Under mild assumptions on the coefficients of both the true and the assumed models, we prove that: (i) the prediction filter, and its gradient with respect to some parameter in the model, forget almost surely their initial condition exponentially fast, and (ii) the extended Markov chain, whose components are the unobserved Markov chain, the observation sequence, the prediction filler, and its gradient, is geometrically ergodic and has a unique invariant probability distribution.