Journal of Applied Mathematics and Stochastic Analysis
Volume 2004 (2004), Issue 4, Pages 371-384
A deterministic discretisation-step upper bound for state estimation via Clark transformations
1Haskayne School of Business, University of Calgary, 2500 University Drive NW, Alberta, Calgary T2N 1N4, Canada
2School of Mathematical Sciences, The University of Adelaide, 5005, SA, Australia
Received 11 November 2003; Revised 2 June 2004
Copyright © 2004 W. P. Malcolm et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.