Linear minimum mean square filter for discrete-time linear systems with multiplicative noise

In this paper we obtain the linear minimum mean square filter for discrete-time linear systems subject to state and measurement multiplicative noises. By using some usual geometric arguments we obtain a Kalman type filter conveniently implementable in a recurrence form. The stationary case is also studied and a proof for the convergence of the associated Lyapunov-like and Riccati-like equations is presented.