Robustness investigation of the numerical approximation of the Wonham filter

The properties of the optimal continuous Markov chain state filtering problem decision given be the linear observations noisy Wiener process, assuming incomplete information about its intensity, are investigated. The uncertainty of the observation system is set by the upper bound of the noise intensity. Numerical implementation of the optimal solution in the statement with complete information provided by the Wonham filter does not guarantee stability. It is shown that the Wonham filter in the statement with uncertainty is robust with respect to the noise intensity if the model parameters do not lead to its divergence. In the general case, the instability of the Euler–Maruyama numerical scheme of the Wonham filter is preserved. Simple heuristic techniques that provide stable approximations of the Wonham filter show the workability for a wider set of parameters. However, in the statement with uncertainty, it is possible to give examples when such heuristic filters show unacceptably low quality. The best solution is provided by discretized filters, approximations of the Wonham filter implemented for a specific model approximating the initial continuous observation system. A series of numerical experiments has shown that these filters have robustness for all sets of parameters. If there are no divergent trajectories among the modeled trajectories of the Wonham filter in the calculations, then the discretized filters lose a little. If there are divergent trajectories, then the gain of discretized filters is absolute.