In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters.
Léandre, Rémi and Mohammed, Salah-Eldin A. "Stochastic Functional Differential Equations on Manifolds." (Sep 2001).