Sparse Wiener Chaos approximations of nonlinear filtering with correlated noise

The nonlinear filtering problem is considered for the time homogeneous diffusion model with correlated noise. A numerical approach is proposed for computing approximations of the unnormalized filtering density (UFD) and the nonlinear filtering. This approach is based on the Wiener chaos expansion (WCE) of the solution of Zakai equation. A Sparse truncation method of WCE is introduced to simplify calculation, and the Sparse truncation error is analyzed. Based on this truncation method, the numerical approximation scheme can substantially reduce the amount of computation and storage, or improve the precision of approximations of the UFD. When r = 1, N = 5 and n = 8, the total number of WCE coefficients will be reduced dramatically, from 1287 to 65.