On extending classical filtering equations

In this paper, we give a direct derivation of the Duncan-Mortensen-Zakai filtering equation, without assuming right continuity of the signal, nor its filtration, and without the usual finite energy condition. As a consequence, the Fujisaki-Kallianpur-Kunita equation is also derived. Our results can be applied to filtering problems in which the signal process has α-stable (α>1) components, and the sensor function is linear.