Projections of SDEs onto Submanifolds

TL;DR

This paper explores three natural, coordinate-free projections of SDEs onto submanifolds, deriving explicit formulas and demonstrating their optimality properties through examples and optimization criteria.

Contribution

It provides a coordinate-free framework for the three projections and establishes their optimality in small-time regimes, expanding on prior work with new formula derivations.

Findings

Ito-vector and Ito-jet projections satisfy weak and mean-square optimality criteria.

Explicit formulas for the projections in ambient coordinates are derived.

Examples show differences among the projections and alternative optimality notions.

Abstract

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.