On the numerical stability of time-discretised state estimation via Clark transformations

In this article 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 a state process through a Brownian motion, and b) the observation of a state process through a Poisson process. For the models just described, one can choose between several different approximate discrete time recursions. However, most of these schemes suffer an inherent instability, that is, their estimated filter probabilities can be negative (with a nonzero probability). We show that there is an exception to this problem, afforded by the so called robust filters due to J. M. C. Clark. It is shown that for the said robust filter, 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 one can choose a priori, a 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.