Filtering of Markov jump processes given composite observations II: Numerical algorithm

The note represents the second, final part of the series initiated by the article Borisov, A., and D. Kazanchyan. 2021. Filtering of Markov jump processes given composite observations I: Exact solution. Informatika i ee primeneniya — Inform. Appl. 15(2):12–19. The authors propose a new numerical algorithm of the optimal state estimation for the Markov jump processes given observable both the counting processes and the diffusion ones with the multiplicative noises. The authors approximate the initial continuous-time estimation problem by a sequence of the corresponding filtering problems given the time-discretized observations. The paper contains the explicit recursive form of the discretized estimate and introduces its one-step precision characteristic along with dependence of the characteristics on the utilized numerical estimation scheme.