*4.4. Numerical Example*

To illustrate the correspondence between the theoretical estimate and its realization along with the performance of the numerical algorithm, we consider the filtering problem for the observation system (1) and (2) with the following parameters: *t* ∈ [0, 1], *N* = 3,

$$
\Lambda = \begin{bmatrix} -1.0 & 0.2 & 0.8 \\ 0.8 & -1.0 & 0.2 \\ 0.2 & 0.8 & -1.0 \end{bmatrix}, \quad \pi = \begin{bmatrix} 0.333 \\ 0.333 \\ 0.334 \end{bmatrix}, \quad f = \begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}, \quad \text{G}\_2 = 4.0,
$$

The specified observation system is the one with state-dependent noise, and the conditions of Corollary 1 hold, so the optimal filter (23) restores the MJP state precisely under available noisy observations. Let us verify this theoretical fact, using the recursive algorithm (37). We choose the analytical approximation of the order *s* = 1 with numerical integration by the simple midpoint rectangle scheme and calculate estimate approximations with decreasing time-discretization step: *h* = 0.01; 0.001; 0.0001; 0.00001. We expect the descent of the estimation error characterized by the MS-criterion S*t*(*h*) = **E** *Xt* − X *t h* 2 2 . To calculate the criterion, we use the Monte–Carlo method over the test sample of the size 1000. Figure 1 presents the corresponding plots of the quality index S*t*(*h*) for various values of *h*.

**Figure 1.** Estimation quality index S*t*(*h*) depending on the time-discretization step *h*.

The determination of the precision order provided by the chosen numerical integration method is out of the scope of this investigation. Nevertheless, one can see the expected decrease of the estimation error when the time-discretization step descends. We appraise this result as a practical confirmation of both the theoretical assertions and numerical algorithm.
