The Variational Approach in Filtering and Correlated Noise
This addresses a foundational limitation in nonlinear filtering theory for systems with shared noise sources, offering a generalization that is incremental but necessary for handling correlated noise scenarios.
The paper tackled the failure of the variational filtering formulation under correlated noise, proving mutual singularity of measures and introducing a conditional variational principle that recovers the original as a special case and yields an explicit characterization for linear correlated noise.
The variational formulation of nonlinear filtering due to Mitter and Newton characterizes the filtering distribution as the unique minimizer of a free energy functional involving the relative entropy with respect to the prior and an expected energy. This formulation rests on an absolute continuity condition between the joint path measure and a product reference measure. We prove that this condition necessarily fails whenever the signal and observation diffusions share a common noise source. Specifically we show that the joint and product measures are mutually singular, so no choice of reference measure can salvage the formulation. We then introduce a conditional variational principle that replaces the prior with a reference measure that preserves the noise correlation structure. This generalization recovers the Mitter--Newton formulation as a special case when the noises are independent, and yields an explicit free energy characterization of the filter in the linear correlated-noise setting.