Abstract
We consider the numerical stability
of discretisation schemes for continuous-time state estimation filters. The dynamical systems
we consider model the indirect observation
of a continuous-time Markov chain. Two candidate
observation models are studied. These models are (a) the observation of the state through a Brownian motion,
and (b) the observation of the state through a Poisson process.
It is shown that for robust filters (via Clark's transformation),
one can ensure nonnegative estimated probabilities by choosing a
maximum grid step to be no greater than a given bound. The
importance of this result is that one can choose an a priori grid step maximum ensuring nonnegative estimated probabilities. In
contrast, no such upper bound is available for the standard
approximation schemes. Further, this upper bound also applies to
the corresponding robust smoothing scheme, in turn ensuring
stability for smoothed state estimates.