Some properties of the nonlinear filter: Markovity and ergodicity

In this paper we first prove, under quite general conditions, that the nonlinear filter and the pair: (signal,filter) are Feller-Markov processes. The state space of the signal is allowed to be non locally compact and the only condition on the observation function, h, is that it be continuous. Our proofs in contrast to those of H. Kunita (1971,1991), L. Stettner (1989) do not depend upon the uniqueness of the solutions to the filtering equations. Indeed, in the generality we consider, the uniqueness of the solutions may not hold. We then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process. These results extend those of Kunita and Stettner, which hold for locally compact state space and bounded h, to our general framework. Finally we show that the recent results of D. Ocone and E. Pardoux (1996) on asymptotic stability of the nonlinear filter, which use the Kunita-Stettner setup, hold for the general situation considered in this paper