  We propose a novel small time approximation for the solution to the Zakai
equation from nonlinear filtering theory. We prove that the unnormalized
filtering density is well described over short time intervals by the solution
of a deterministic partial differential equation of Kolmogorov type; the
observation process appears in a pathwise manner through the degenerate
component of the Kolmogorov's type operator. The rate of convergence of the
approximation is of order one in the lenght of the interval. Our approach
combines ideas from Wong-Zakai-type results and Wiener chaos approximations for
the solution to the Zakai equation. The proof of our main theorem relies on the
well-known Feynman-Kac representation for the unnormalized filtering density
and careful estimates which lead to completely explicit bounds.
