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

The authors consider the nonlinear filtering model with additive white nois e taken to be the identity map on L-2[0,T] with standard Gauss measure ther eon. Using a representation result for maps which are continuous in a local ly convex topology generated by seminorms of Hilbert-Schmidt operators on t he Hilbert space, the authors show that the filter map can be written as th e composition of a continuous nonlinear map (which does not depend on the o bservation) with a linear Hilbert-Schmidt operator acting on the observatio n. In particular, this result gives a direct proof of existence of approxim ation of nonlinear filters in terms of Volterra polynomials.