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We define two new notions of projection of a stochastic differential equation(SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allowsone to systematically develop low dimensional approximations to highdimensional SDEs using differential geometric techniques. The approachgeneralizes the notion of projecting a vector field onto a submanifold in orderto derive approximations to ordinary differential equations, and improves theprevious Stratonovich projection method by adding optimality analysis andresults. Indeed, just as in the case of ordinary projection, our definitions ofprojection are based on optimality arguments and give in a well-defined sense"optimal" approximations to the original SDE in the mean-square sense. We alsoshow that the Stratonovich projection satisfies an optimality criterion that ismore ad hoc and less appealing than the criteria satisfied by the Itoprojections we introduce. As an application we consider approximating thesolution of the non-linear filtering problem with a Gaussian distribution andshow how the newly introduced Ito projections lead to optimal approximations inthe Gaussian family and briefly discuss the optimal approximation for moregeneral families of distribution. We perform a numerical comparison of ouroptimally approximated filter with the classical Extended Kalman Filter todemonstrate the efficacy of the approach.

Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections