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 $\sqrt{\varepsilon}$. 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. 
Peter Imkeller, N. Sri Namachchivaya, Nicolas Perkowski, Hoong C.   Yeong
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