Filtering problem with nonlinear observations and drift terms equal to gradient vector field plus affine vector field

For all known finite dimensional filters, one always needs the condition that the observation terms are degree one polynomial. On the other hand, in many practical examples, e.g. tracking problem, the observation terms may be nonlinear. In this paper, we solve filtering problems with nonlinear observation terms and drift term equal to gradient vector field plus affine vector field, which includes Kalman-Bucy filter, Benes filter and Yau filter as special cases.