Exact finite-dimensional filters for certain diffusions with nonlinear drift

Let and be independent Wiener processes, and consider the task of estimating a diffusion solving the stochastic DE dx t =f(x t )dt+dw t on the basis of noisy observations defined bydy t =x t dt+db t . This problem is governed by the filtering equation for the unnormalized conditional density with A * the forwarded operator Theorem: if then the fundamental solution of the filtering equation can be written explicity in terms of a small number of statistics satisfying a matrixvector equation. The Lie algebraic interpretation of this result is studied and described. Extensions to many dimensions and applications to optimal stochastic control readily follow.