 
Summary: Exponential decay rate of the filter's
dependence on the initial distribution
Rami Atar
January 15, 2009 (Revised May 7, 2009)
Abstract: We review how tools from multiplicative ergodic theory and the theory of
positive operators are used in the analysis of exponential stability of the optimal nonlinear
filter. Particularly, in the case of finite state, we relate the filter sensitivity to perturbations
in its initial data to the Lyapunov spectral gap associated with the filtering equation, and,
in a general setting, use Hilbert's metric and Birkhoff's contraction coefficient to estimate
the decay rate of the error.
1. Introduction
The problem of stability of the nonlinear filter arises in the following practical context. If
the transition law of a given Markov process is known, but its initial law is not available,
under what conditions does one not lose optimality of the filter when initializing it with an
arbitrary (thus wrong) initial data, in the limit when time tends to infinity? This question, first
posed by Ocone and Pardoux [35] and Delyon and Zeitouni [20], has attracted much attention.
This paper focuses on the exponential rate of decay of the error made by wrong initialization,
and reviews results that relate this quantity to multiplicative ergodic theory (MET) on one
hand, and to Hilbert's metric and Birkhoff's contraction coefficient on the other hand. MET
is instrumental in establishing that, in a finite state setting, the decay rate is deterministic
