Projections of SDEs onto Submanifolds

In [ABF19] the authors define three projections of Rd-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Ito-vector and Ito-jet projections. In this paper, after a brief survey of SDEs on manifolds, we begin by giving these projections a natural, coordinate-free description, each in terms of a specific representation of manifold-valued SDEs. We proceed by deriving formulae for the three projections in ambient $\mathbb R^d$-coordinates. We use these to show that the Ito-vector and Ito-jet projections satisfy respectively a weak and mean-square optimality criterion for small t: this is achieved by solving constrained optimisation problems. These results confirm, but do not rely on the approach taken in [ABF19], which is formulated in terms of weak and strong Ito-Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality.