  We present the two new notions of projection of a stochastic differential
equation (SDE) onto a submanifold, as developed in Armstrong, Brigo e Rossi
Ferrucci (2019, 2018): the Ito-vector and Ito-jet projections. This allows one
to systematically and optimally develop low dimensional approximations to high
dimensional SDEs using differential geometric techniques. Our new projections
are based on optimality arguments and yield a well-defined ``optimal''
approximation to the original SDE in the mean-square sense. We also show that
the earlier Stratonovich projection satisfies an optimality criterion that is
more ad hoc and less natural than the criteria satisfied by the new
projections. As an application, we consider approximating the solution of the
non-linear filtering problem within a given manifold of densities, using either
the Hellinger or $L^2$ direct metrics and related Information Geometry
structures on the space of densities. The Stratonovich projection had yielded
the projection filters studied in Brigo, Hanzon and Le Gland (1998, 1999),
while the new projections lead to the optimal projection filters. The optimal
projection filters have been introduced in Armstrong, Brigo e Rossi Ferrucci
(2019), where numerical examples for the Gaussian case are given and where they
are compared to more traditional nonlinear filters.
