Filtering of Markov jump processes given composite observations I: Exact solution

The first part of the series is devoted to the optimal filtering of the finite-state Markov jump processes (MJP) given the ensemble of the diffusion and counting observations. The noise intensity in the observable diffusion depends on the estimated MJP state. The special equivalent observation transformation converts them into the collection of the diffusion process of unit intensity, counting processes, and indirect measurements performed at some nonrandom discrete instants. The considered filtering estimate is expressed as a solution to the discrete-continuous stochastic differential system with the transformed observations on the right-hand side. The identifiability condition, under which MJP state can be reconstructed from indirect noisy observations precisely, is presented.