Projections of SDEs onto submanifolds

In Armstrong et al. (Proc Lond Math Soc (3) 119(1):176-213, 2019) the authors define three projections of $R^d$ -valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Itô-vector and Itô-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 $R^d$ -coordinates. We use these to show that the Itô-vector and Itô-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 Armstrong et al. (Proc Lond Math Soc (3) 119(1):176-213, 2019), which is formulated in terms of weak and strong Itô-Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality.