Optimal filtering and control of linear systems with Markov perturbations

The stochastic model under consideration is a linear jump diffusion process X for which the coefficients and the jump processes depend on a Markov chain, Z, with finite state space. First we study the optimal filtering and control problem for these systems with non-gaussian initial conditions for noisy observations of the state X and perfect observations of Z. Under technical assumptions it is proved that the conditional characteristic function of X is parametrically determined by a finite set of statistics. Next, we derive a new sufficient condition which ensures the existence and the uniqueness of the solution of the nonlinear stochastic differential equations satisfied by the output of the filter, extending the result of Haussmann (1987). We study a quadratic control problem and show that the separation principle holds. We give the form of the controller, which can be explicitly calculated in term of the optimal filter. The gain of the controller depends on a system of modified, coupled Riccati equations. The existence of its solution is proved. Our results widen the class of linear systems for which the separation principle holds