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

Abstract

Given a stochastic differential equation (SDE) inRnwhose solution is constrained to lie in some manifoldM⊂Rn, we identify a class of numerical schemes for the SDE whose iterates remain close toMto high order. These schemes approximate a geometrically invariant scheme, which gives perfect solutions for any SDE that is diffeomorphic ton-dimensional Brownian motion. Unlike projection-based methods, they may be implemented without explicit knowledge ofM. They can even be implemented if the solution merely remains close toM, 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.