Exact solution to the Zakai equation for certain diffusions

An exact expression for the unnormalized conditional density is derived for certain diffusions with nonlinear drift terms. This corresponds to the exact solution of the Zakai equation for certain nonlinear estimation problems. This represents a generalization of both the Kalman-Bucy filter and the Bene?? filter within a unified framework. Moreover, the new filter can handle a large class of nonlinear estimation problems, including diffusions with the generalized Maxwell-Boltzmann distribution, that cannot be accommodated by either the Kalman-Bucy filter or the Bene?? filter.