Dimensional reduction in nonlinear filtering: A homogenization approach

We propose a homogenized filter for multiscale signals, which allows us to reduce the dimension of the system. We prove that the nonlinear filter converges to our homogenized filter with rate $\sqrt{\epsilon }$. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and by probabilistically representing the correction terms with the help of BDSDEs.