Geometry-preserving Numerical Scheme for Riemannian Stochastic Differential Equations

Xi Wang; Victor Solo

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arXiv · 2504.12631

Geometry-preserving Numerical Scheme for Riemannian Stochastic Differential Equations

Xi Wang·Victor Solo

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Abstract

Stochastic differential equations (SDEs) on Riemannian manifolds have numerous applications in system identification and control. However, geometry-preserving numerical methods for simulating Riemannian SDEs remain relatively underdeveloped. In this paper, we propose the Exponential Euler-Maruyama (Exp-EM) scheme for approximating solutions of SDEs on Riemannian manifolds. The Exp-EM scheme is both geometry-preserving and computationally tractable. We establish a strong convergence rate of $\mathcal{O}(\delta^{\frac{1 - \epsilon}{2}})$ for the Exp-EM scheme, which extends previous results obtained for specific manifolds to a more general setting. Numerical simulations are provided to illustrate our theoretical findings.

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Xi Wang, Victor Solo. 2025-04-17. Geometry-preserving Numerical Scheme for Riemannian Stochastic Differential Equations. https://arxiv.org/abs/2504.12631

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