Discrete time nonlinear filters with informative observations are stable
Ramon Van Handel, Princeton University
Abstract
The nonlinear filter associated with the discrete time signal-observation
model $(X_k,Y_k)$ is known to forget its initial condition as $ktoinfty$
regardless of the observation structure when the signal possesses
sufficiently strong ergodic properties. Conversely, it stands to reason
that if the observations are sufficiently informative, then the nonlinear
filter should forget its initial condition regardless of any properties of
the signal. We show that for observations of additive type
$Y_k=h(X_k)+xi_k$ with invertible observation function $h$ (under mild
regularity assumptions on $h$ and on the distribution of the noise
$xi_k$), the filter is indeed stable in a weak sense without any
assumptions at all on the signal process. If the signal satisfies a
uniform continuity assumption, weak stability can be strengthened to
stability in total variation.
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