A white noise version of the Girsanov theorem

Let M be a nonlinear transformation on a separable Hilbert space with range in H. Let μ denote a standard Gauss measure on H. It is shown that, under suitable conditions on M, there exists an exponential transformation L (completely characterized by M) on H such that d η=Ldμ defines a finitely additive or cylindrical probability measure on H under which (I-M)(.) is white noise. This is the white noise version of the Girsanov theorem. The nonlinear filtering model is considered, and the Radon-Nikodym derivative of the cylindrical measure induced by the observation process on H is interpreted, showing that it has a pathwise characterization in terms of the nonlinear filter map. It is then shown that if the signal process is the solution to a nonlinear differential equation with a white noise input, then the innovation process is white noise under the cylindrical measure induced by the observation process and the innovations process is related to the observation process by a continuous, causally invertible map