Curved schemes for stochastic differential equations on, or near, manifolds

Given a stochastic differential equation (SDE) in Rn whose solution is constrained to lie in some manifold M Rn, we identify a class of numerical schemes for the SDE whose iterates remain close to M to high order. These schemes approximate a geometrically invariant scheme, which gives perfect solutions for any SDE that is diffeomorphic to n-dimensional Brownian motion. Unlike projection-based methods, they may be implemented without explicit knowledge of M. They can even be implemented if the solution merely remains close to M, without being exactly confined to it. Our approach does not require simulating any iterated Itô integrals beyond those needed to implement the Euler-Maryuama scheme. We prove that the schemes converge under a standard set of assumptions, and illustrate their geometric advantages in a variety of numerical contexts, including Monte Carlo simulation of the Riemannian Langevin equation.