Markov property and ergodicity of the nonlinear filter

In this paper we rst prove, under quite general conditions, that the nonlin ear filter and the pair (signal, filter) are Feller-Markov processes. The s tate space of the signal is allowed to be nonlocally compact and the observ ation function h can be unbounded. Our proofs, in contrast to those of Kuni ta [J. Multivariate Anal., 1 (1971), pp. 365-393; Spatial Stochastic Proces ses, Birkhauser, 1991, pp. 233-256] and Stettner [Stochastic Differential E quations, Springer-Verlag, 1989, pp. 279-292], do not depend upon the uniqu eness of the solutions to the filtering equations. We then obtain condition s for existence and uniqueness of invariant measures for the nonlinear filt er 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 f ramework. Finally we show that the recent results of Ocone and Pardoux [SIA M J. Control Optim., 34 (1996), pp. 226-243] on asymptotic stability of the nonlinear filter, which use the Kunita Stettner setup, hold for the genera l situation considered in this paper.