$\mathcal{L}_1$-optimal filtering of Markov jump processes. II. Numerical analysis of particular realizations schemes

Part II of the paper deals with particular numerical schemes used to realize the filtering algorithm for Markov jump processes by indirect observations corrupted by Wiener noises. The orders of accuracy of these numerical schemes are determined. The cases of additive and multiplicative noises in observations are investigated separately: as shown below, the same schemes in these cases have different accuracy. For observations with additive noises, schemes of orders $\frac{1}{2}$ , $1$ and $2$ are proposed; for observations with multiplicative noises, schemes of orders $1$ and $2$. The theoretical results are illustrated with numerical examples.