A representation result for nonlinear filter maps in a white noise framework

The authors consider the nonlinear filtering model with additive white noise to be the identity map on L ₂[0,T] with standard Gauss measure thereon. Using a representation result for maps which are continuous in a locally convex topology generated by seminorms of Hilbert-Schmidt operators on the Hilbert space, the authors show that the filter map can be written as the composition of a continuous nonlinear map (which does not depend on the observation) with a linear Hilbert-Schmidt operator acting on the observation. In particular, this result gives a direct proof of existence of approximation of nonlinear filters in terms of Volterra polynomials.