Curved Schemes for SDEs on Manifolds

Given a stochastic differential equation (SDE) in $\mathbb{R}^n$ whose solution is constrained to lie in some manifold $M \subset \mathbb{R}^n$, we propose a class of numerical schemes for the SDE whose iterates remain close to $M$ to high order. Our schemes are geometrically invariant, and can be chosen to give perfect solutions for any SDE which is diffeomorphic to $n$-dimensional Brownian motion. Unlike projection-based methods, our schemes may be implemented without explicit knowledge of M. Our approach does not require simulating any iterated It\^{o} interals beyond those needed to implement the Euler--Maryuama scheme. We prove that the schemes converge under a standard set of assumptions, and illustrate their practical advantages by considering a stochastic version of the Kepler problem.