Suppose I have a Riemannian manifold $(M,g)$ embedded in $\mathbb{R}^n$ and I can observe Brownian motions on it at some reasonable timescale; that is, I can observe $\hat{X}_{i,t}$ as a sequence of discrete positions over time $\hat{X}_{i,t}=\{\hat{X}_{i,0},\hat{X}_{i,\delta}, \ldots\}$ i.e. each being samples at sampling frequency $\delta$ on the continuous path of the $i$the Brownian motion trajectory.
We know that any of these observed motions were generated by an SDE describing an Ito diffusion process: $$ dX_t^i = \sigma^i_j(X_t)dB^j_t +\frac{1}{2}b^i(X_t)dt $$ (e.g. see Hsu, A Brief Introduction to Brownian Motion on a Riemannian Manifold), where $\sigma$ and $b$ depend on $g$, and here $i$ is referring to a component of the Brownian motion.
Is there a way to determine $\sigma$ and $b$ from these measurements? Or, even better, to determine $g$?