Markov property and ergodicity of the nonlinear filter

In this paper we rst 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 nonlocally compact and the observation function h can be unbounded. Our proofs, in contrast to those of Kunita [J. Multivariate Anal., 1 (1971), pp. 365-393; Spatial Stochastic Processes, Birkhauser, 1991, pp. 233-256] and Stettner [Stochastic Differential Equations, Springer-Verlag, 1989, pp. 279-292], do not depend upon the uniqueness of the solutions to the filtering equations. 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 Ocone and Pardoux [SIAM J. Control Optim., 34 (1996), pp. 226-243] on asymptotic stability of the nonlinear filter, which use the Kunita Stettner setup, hold for the general situation considered in this paper.