THE STABILITY OF CONDITIONAL MARKOV PROCESSES ANDMARKOV CHAINS IN RANDOM ENVIRONMENTS

摘要

We consider a discrete time hidden Markov model where the signal is astationary Markov chain. When conditioned on the observations, the signal isa Markov chain in a random environment under the conditional measure. It isshown that this conditional signal is weakly ergodic when the signal is ergodicand the observations are nondegenerate. This permits a delicate exchange ofthe intersection and supremum of a-fields, which is key for the stability ofthe nonlinear filter and partially resolves a long-standing gap in the proof ofa result of Kunita [J. Multivariate Anal. 1 (1971) 365-393]. A similar resultis obtained also in the continuous time setting. The proofs are based on anergodic theorem for Markov chains in random environments in a general statespace.