Geometry-preserving numerical scheme for Riemannian stochastic differential equations

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 O ( δ 1 − ϵ 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.