A note on exponential stability of the nonlinear filter for denumerable Markov chains

Asymptotic stability of the optimal filter with respect to its initial conditions is investigated in this paper. Under the assumption that the observation function is one to one and the observation noise is sufficiently small, it is shown that exponential stability of the nonlinear filter holds for a large class of denumerable Markov chains, including all finite Markov chains. Throughout this paper, ergodicity of the signal process is not assumed.