Numerical schemes of Markov jump process filtering given discretized observations I: Accuracy characteristics

The note is the initial in the series of the papers devoted to the numerical realization of the optimal state filtering of Markov jump processes given the indirect observations corrupted by the additive and/or multiplicative Wiener noises. This problem is solved by the time discretization of the observations with their subsequent processing. Both the optimal and suboptimal estimations are expressed in terms of multiple integrals of the Gaussian densities with some mixing distributions. In the article, the author presents the investigation of various numerical integration *schemes' influence on the accuracy of the approximating estimates. The problem turns into the characterization of distance between stochastic sequences generated by some recursions. The paper introduces a pseudometric describing the distance and presents a proposition determining the influence of the characteristic on both the local and global accuracy of the filtering estimate approximation.