Wiener Chaos and Nonlinear Filtering

The paper discusses two algorithms for solving the Zakai equation in the time-homogeneous diffusion filtering model with possible correlation between the state process and the observation noise. Both algorithms rely on the Cameron-Martin version of the Wiener chaos expansion, so that the approximate filter is a finite linear combination of the chaos elements generated by the observation process. The coefficients in the expansion depend only on the deterministic dynamics of the state and observation processes. For real-time applications, computing the coefficients in advance improves the performance of the algorithms in comparison with most other existing methods of nonlinear filtering. The paper summarizes the main existing results about these Wiener chaos algorithms and resolves some open questions concerning the convergence of the algorithms in the noise-correlated setting. The presentation includes the necessary background on the Wiener chaos and optimal nonlinear filtering.