Controllable Markov Jump Processes. I. Optimum Filtering Based on Complex Observations

Abstract

The first part of the paper is devoted to justifying the possibility of the correct description of a controllable stochastic observation system. The state is a Markov jump process, and the observations are a combination of continuous, discrete, and counting processes. A martingale problem of the system under study is solved: it is shown that there exists a canonical probability space with filtration such that under any admissible control, this system is stochastic differential with the martingales on the right-hand side. Further, for this system there exists a solution of the optimal in the mean square sense filtering problem given the compound observations. The filtering estimate is presented in the form of a continuous-discrete stochastic system with the martingales on the right-hand side. The article contains a description of a numerical algorithm implementing both process modeling in the considered observation system and the proposed solution for the filtering problem.