$\mathcal {L}_1$-optimal filtering of Markov jump processes. III. Identification of system parameters

The present paper is a continuation of the series of articles [1, 2] and is devoted to solving the problem of estimating the parameters of hidden Markov models. The hidden state is a homogeneous Markov jump process with a finite set of states. The available observations are indirect and contain Wiener processes whose intensities are different and depend on the hidden state. Both the intensity matrix of Markov state transitions and the drift and diffusion parameters of the observations are subject to estimation. For identification, an iterative algorithm based on smoothing the state of the system based on observations over a fixed time interval is proposed. Then, according to these estimates, the parameters are reconstructed. The paper describes in detail all the numerical schemes for estimating the state and for identifying the parameters. A set of illustrative numerical examples is presented, demonstrating the high quality of the proposed identification estimates.