Extend the conditional variational formulation to nonlinear correlated-noise diffusions

Extend the conditional variational principle for filtering with correlated signal and observation noise—which replaces the prior by a conditional reference measure that preserves the noise coupling and yields a free-energy characterization of the filter—from the linear diffusion case to general nonlinear diffusion models with shared Brownian noise, by formulating an appropriate free-energy functional and proving that the nonlinear filtering distribution is its unique minimizer.

Background

The classical Mitter–Newton variational formulation characterizes the filtering distribution as the unique minimizer of a free-energy functional but relies on absolute continuity between the joint path measure and a product reference measure. The authors prove that this fails fundamentally when signal and observation diffusions share a common noise source, establishing mutual singularity of the joint and product measures.

To address correlated noise, the article introduces a conditional variational principle that replaces the prior with a conditional reference measure preserving the noise correlation structure. This generalization recovers the Mitter–Newton formulation in the independent-noise case and provides an explicit free-energy characterization of the filter in the linear correlated-noise setting.

The remaining challenge is to lift this conditional variational characterization from linear diffusions to general nonlinear diffusion models with correlated noise.