Poisson's equation in nonlinear filtering

Understand

The aim of this paper is to provide a variational interpretation of the nonlinear filter in continuous time.

• A time-stepping procedure is introduced, consisting of successive minimization problems in the space of probability densities.
• The weak form of the nonlinear filter is derived via analysis of the first-order optimality conditions for these problems.
• The derivation shows the nonlinear filter dynamics may be regarded as a gradient flow, or a steepest descent, for a certain energy functional with respect to the Kullback-Leibler divergence.