Tangent Space Parametrization for Stochastic Differential Equations on SO(n)

Abstract

In this paper, we study the numerical simulation of stochastic differential equations (SDEs) on the special orthogonal Lie group $\text{SO}(n)$. We propose a geometry-preserving numerical scheme based on the stochastic tangent space parametrization (S-TaSP) method for state-dependent multiplicative SDEs on $\text{SO}(n)$. The convergence analysis of the S-TaSP scheme establishes a strong convergence order of $\mathcal{O}(\hat{I}{´}^{\frac{1-\hat{I}\mu }{2}})$, which matches the convergence order of the previous stochastic Lie Euler-Maruyama scheme while avoiding the computational cost of the exponential map. Numerical simulation illustrates the theoretical results.