Appl Math Optim 40:259–272 (1999)
1999 Springer-Verlag New York Inc.
Filtering with Discrete State Observations
and R. J. Elliott
Laboratoire des Signaux et Syst`emes, C.N.R.S. – E.S.E.,
Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France
Department of Mathematical Sciences, University of Alberta,
Edmonton, Alberta, Canada T6G 2G1
Abstract. The problem of estimating a ﬁnite state Markov chain observed via a
process on the same state space is discussed. Optimal solutions are given for both
the “weak” and “strong” formulations of the problem. The “weak” formulation
proceeds using a reference probability and a measure change for the Markov chain.
The “strong” formulation considers an observation process related to perturbations
of the counting processes associated with the Markov chain. In this case the “small
noise” convergence is investigated.
Key Words. Filtering, Markov chain, Counting process, Girsanov’s theorem.
AMS Classiﬁcation. 60J27, 93E11.
The problem of estimating a ﬁnite state continuous-time Markov chain observed in
Gaussian noise is well known: its solution is given by the Wonham ﬁlter. With the
exception of the results in , the related problem of estimating such a ﬁnite state
Markov chain observed in a second Markov chain does not appear to have been stud-
ied. The optimal ﬁlter estimate is derived in this paper in both “weak” and “strong”
This work was supported by a NATO Collaborative Research Grant (Ref. CRG 971001) and NSERC