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

W. P. Malcolm,1 R. J. Elliott,1 and J. van der Hoek2

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.