Pathwise Nonlinear Filtering on Abstract Wiener Spaces

Abstract

The nonlinear filtering problem is studied for models where the samples of the signal and the noise are elements of some general abstract Wiener space. The signal is allowed to depend on the noise and the optimal filter is expressed as an explicit functional of the observed sample (trajectory). It is shown that this functional satisfies the Zakai equation. As a necessary technical tool, a class of shift transformations on the Wiener space is studied and an analog of Cameron-Martin-Girsanov's theorem is obtained.