About filtering problem of diffusion processes

The filtering problem of nonlinear one-dimensional diffusion processes is considered. The structures of observable and nonobservable processes are found. It is shown, that solution of the optimal filtering problem can be reduced to solution of the filtering problem for the case when a nonobservable process has a simpler structure and an observable process is the Wiener process with a random smooth trend. The equation connecting a conditional expectation for the initial filtering problem with a nonnormalized filtering density for the reduced filtering problem is obtained.