Exact Filtering in Semi‐Markov Jumping System

The classical hidden linear Gaussian system allows one to use the classical Kalman filter, which calculates some distributions of interest with linear complexity in number of observations. However, such calculations become impossible when adding a Markov jump process. The aim of the paper is to propose two new hidden models with Markov and semi‐Markov jump processes in which the exact computation of the Kalman filter is feasible with linear complexity in number of observations.