I recently saw the following characterization of Brownian motion on an embedded manifold: Let $M$ be a manifold embedded in $\mathbb{R}^n$. Consider some point $p\in M$, for $\vec{x} \in \mathbb{R}^n$ let $P_i(\vec{x})$ be the projection of the $i$th coordinate of $\vec{x}$ onto $T_p M$. Now, Brownian motion on $M$ can be characterized as the following Stratovich SDE: $$dX_t = P_i(X_t) \circ dW^i_t$$ where the Einstein summation convention is used.
I'm trying to understand a discrete approximation. In particular, I want to compute and plot this representation on something simple (ex $S^2$), but because it's not in Ito form, using something like Euler Maruyama won't work.
I'm wondering, can I just blindly use the Stratovich SDE to Ito SDE formula (given on Wikipedia or in Oksendal) to convert this to an Ito SDE, and just perform Euler-Maruyama over $\mathbb{R}^n$ to approximate the process? Or is there a subtlety about being on a manifold that I'm missing? I've searched Google, the Arxiv, and Hsu's book for an Ito SDE that represents Brownian motion extrinsically for an embedded manifold. Why can't I find it? Did I not look hard enough, or is there something about manifolds that doesn't allow for an extrinsic Ito SDE description of Brownian motion?