**5. Conclusions**

In this paper, we investigated the optimal filtering problem of the MJP states, given the indirect noisy continuous-time observations. The observation noise intensity was a function of the estimated state, so it was impossible to apply the classic Wonham filter to this observation system. To overcome this obstacle, we suggested an observation transform. On the one hand, the transformed observations remained to be equivalent to the original one from the informational point of view. On the other hand, the "new" observations allowed to apply the effective stochastic analysis framework to process them. We derived the optimal filtering estimate theoretically as a unique strong solution to some discrete–continuous stochastic differential system. The transformed observations included derivative of the quadratic characteristics, i.e., the result of some limit passage in the stochastic settings. Hence, the subsequent numerical realization of the filtering became challenging. We proposed to approximate the initial continuous-time filtering problem by a sequence of the optimal ones given the time-discretized observations. We also involved numerical integration schemes to calculate the integrals included in the estimation formula. We prove assertions, characterizing the accuracy of the numerical approximation of the filtering estimate, i.e., the distance between the calculated approximation and optimal discrete-time filtering estimate. The accuracy depended on the observation system parameters, time discretization step, a threshold of state transition number during the time step, and the chosen scheme of the numerical integration. We suggested the whole class of numerical filtering algorithms. In each case, one could choose any specific algorithm individually, taking into account characteristics of the concrete observation system, accuracy requirements, and available computing resources.

We do not consider the presented investigations as completed. First, the characterization of the distance between the initial optimal continuous-time filtering estimate and its proposed approximation is still an open problem. Second, we can use the theoretical solution to the MJP filtering problem as a base of numerical schemes for the diffusion process filtering, given the observations with state-dependent noise. Third, the obtained optimal filtering estimate looks a springboard for a solution to the optimal stochastic control of the Markov jump processes, given both the counting and diffusion observations with state-dependent noise. All of this research is in progress.

**Author Contributions:** Conceptualization, A.B., I.S.; methodology, A.B.; formal analysis and investigation, A.B., I.S.; writing—original draft preparation, A.B.; writing—review and editing, I.S.; supervision, I.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
