Filtering of states and parameters of special Markov jump processes via indirect perfect observations

The paper investigates the problem of optimal state filtering of a class of stochastic differential observation systems. The state to be estimated consists of two compound components. The first is a finite-state Markov jump process. The second component changes synchronously with the first one and, given a fixed first component, forms a sequence of independent vectors. The available statistical information includes the known functions of the estimated state observed without noise. The problem is to construct the conditional distribution of the system's state given the available observations. In observation systems with degenerate noise, it is impossible to apply standard filtering techniques, which typically involve reducing the observations to a combination of Wiener and Poisson processes using a suitable Girsanov measure transform. The conditional distribution of the state can be represented using a recursively linked sequence of ordinary differential and difference equations.