Projections of SDEs onto submanifolds (Q6138806)

As an extension to [\textit{J. Armstrong} et al., Proc. Lond. Math. Soc. (3) 119, No. 1, 176--213 (2019; Zbl 1421.58008)], where three projections of 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 manifoldvalued SDEs. We proceed by deriving formulae for the three projections in ambient Rd -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 [loc. cit.], 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.