Dimensional reduction in nonlinear filtering

The theory of nonlinear filtering forms the framework of many data assimilation problems. When the rates of change of different variables differ by orders of magnitude, efficient data assimilation can be accomplished by constructing nonlinear filtering equations for the coarse-grained signal. We consider the conditional law of a signal given the observations in a multi-scale context. In particular, we study how scaling interacts with filtering via stochastic averaging. This is an extension of our previous work (Park et al 2008 Stoch. Dyn. 8 543-60) where the observation process depended only on the fast variable, so the filter became independent of the observation in the limit. Here, we investigate a more realistic setting in which the observation depends on both the slow and the fast variables.