Numerical approximation of diffusions in using normal charts of a Riemannian manifold

We consider a numerical scheme approximation for a diffusion on a Riemannian manifold using a normal chart approach and prove that it coincides with the modified Milstein scheme introduced in [A.B. Cruzeiro, P. Malliavin, A. Thalmaier, Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 481–486].