Curved Schemes for SDEs on Manifolds

John Armstrong; Tim King

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arXiv · 2009.10113

Curved Schemes for SDEs on Manifolds

John Armstrong·Tim King

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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.

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John Armstrong, Tim King. 2020-09-21. Curved Schemes for SDEs on Manifolds. https://arxiv.org/abs/2009.10113

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