  We define two new notions of projection of a stochastic differential equation
(SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows
one to systematically develop low dimensional approximations to high
dimensional SDEs using differential geometric techniques. The approach
generalizes the notion of projecting a vector field onto a submanifold in order
to derive approximations to ordinary differential equations, and improves the
previous Stratonovich projection method by adding optimality analysis and
results. Indeed, just as in the case of ordinary projection, our definitions of
projection are based on optimality arguments and give in a well-defined sense
"optimal" approximations to the original SDE in the mean-square sense. We also
show that the Stratonovich projection satisfies an optimality criterion that is
more ad hoc and less appealing than the criteria satisfied by the Ito
projections we introduce. As an application we consider approximating the
solution of the non-linear filtering problem with a Gaussian distribution and
show how the newly introduced Ito projections lead to optimal approximations in
the Gaussian family and briefly discuss the optimal approximation for more
general families of distribution. We perform a numerical comparison of our
optimally approximated filter with the classical Extended Kalman Filter to
demonstrate the efficacy of the approach.
