In the existing `direct¿ white noise theory of nonlinear filtering, the state process is still modelled as a Markov process satisfying an Itô stochastic differential equation, while a `finitely additive¿ white noise is used to model the observation noise. We remove this asymmetry by modelling the state process as the solution of a (stochastic) differential equation with a `finitely additive¿ white noise as the input. This enables us to introduce correlation between the state and observation noises, and to obtain robust nonlinear filtering equations in the correlated noise case.
Bagchi, A., & Karandikar, R. (1994). White noise theory of robust nonlinear filtering with correlated state and observation noises. Systems and control letters, 23(23), 137-148. https://doi.org/10.1016/0167-6911(94)90043-4