Numerical schemes of Markov jump process filtering given discretized observations III: Multiplicative noises case

The paper presents the final part of investigations initialized in the papers Borisov, A. 2019. Numerical schemes of Markov jump process filtering given discretized observations I: Accuracy characteristics. Inform. Appl. 13(4):68–75 and Borisov, A. 2020. Numerical schemes of Markov jump process filtering given discretized observations II: Multiplicative noises case. Inform. Appl. 14(1):17–23. Relying on the theoretical results, this paper presents a numerical algorithm of the state filtering of homogeneous Markov jump processes (MJP) given indirect noisy continuous time observations discretized by time. The class of observation systems under consideration is restricted by ones with multiplicative noises: any additive payload component is absent in the observable signal, but the observation noise intensity is a function of the MJP state under estimation. To calculate the integrals in the estimate, the author uses the composite midpoint rule of the precision order $3$, along with the composite midpoint rule for triangles of the precision order $4$. The constructed numerical algorithms of filtering have the final precision of the orders $1$ and $2$.