Extrinsic projection of Itˆo SDEs on submanifolds with applications to non-linear filtering

. We define the notion of the extrinsic Itˆo projection of a stochastic differential equation (SDE) on a submanifold. This allows one to systematically develop low dimensional approximations to high dimensional SDEs in a differential geometric setting. We consider the example of approximating the non-linear filtering problem with a Gaussian distribution and show how the Itˆo projection leads to improved approximations in the Gaussian family. We briefly discuss the approximations for more general families of distribution. We perform a numerical compar-ison of our projection filters with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

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