The Annals of Probability
- Ann. Probab.
- Volume 37, Number 5 (2009), 1876-1925.
The stability of conditional Markov processes and Markov chains in random environments
We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of σ-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365–393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space.
Ann. Probab., Volume 37, Number 5 (2009), 1876-1925.
First available in Project Euclid: 21 September 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 93E11: Filtering [See also 60G35]
Secondary: 60J05: Discrete-time Markov processes on general state spaces 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 93E15: Stochastic stability
van Handel, Ramon. The stability of conditional Markov processes and Markov chains in random environments. Ann. Probab. 37 (2009), no. 5, 1876--1925. doi:10.1214/08-AOP448. https://projecteuclid.org/euclid.aop/1253539859