Abstract:

The problem under consideration is the filtering of Gaussian noise observations of a linear differential system driven by both Brownian motion and a Markov process switching in continuous time at a constant rate $\lambda$ with state space $(-1, +1)$ usually referred to as a random telegraph process. The algorithms compared are: (1) The Interacting Multiple Model (IMM) algorithm. (2) Differential equations driven by the innovation process for the mean and variance of the state and switching level derived from a representation for the posterior density of the joint process, which is in turn obtained from the fundamental filtering theorem for semi-martingales with Gaussian observation noise. (3) A filter obtained by replacing the Markov switching process by a Gaussian process with equivalent second order properties. This gives rise to a Kalman-Bucy filter.