Curved Schemes for SDEs on Manifolds

TL;DR

This paper introduces a new class of geometric numerical schemes for SDEs constrained to manifolds, which remain close to the manifold with high accuracy and do not require explicit manifold knowledge.

Contribution

The paper presents manifold-invariant numerical schemes for SDEs that avoid explicit projection and iterated Itô integrals, improving accuracy and implementation simplicity.

Findings

Schemes achieve high-order accuracy near manifolds.

Methods are invariant under geometric transformations.

Practical advantages demonstrated on stochastic Kepler problem.

Abstract

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.