Just a question about Infinitesimal generator of the Brownian motion on a sphere I do not see how they got:
$$d\theta_t= \sin\phi_t \circ dB_t^{1}- \cos\phi_t \circ dB^{2}_{t}, \qquad d\phi_t= \cot\theta_t[\cos\phi_t \circ dB_t^{1}+\sin\phi_t\circ dB_t^{2}]-dB_t^{3}$$ And then back to Ito formulation:
$$d\theta_t= \frac{1}{2}\cot\theta_t +\sin\phi_tdB_t^{1}- \cos\phi_t dB^{2}_{t}, \qquad d\phi_t= \cot\theta_t[\cos\phi_tdB_t^{1}+\sin\phi_t dB_t^{2}]-dB_t^{3}$$
I did compute $\frac{\partial x}{\partial \theta}$, $\frac{\partial x}{\partial\phi}$ and compared with the formula for Stratonovich cross product ($dX_t= X_t \circ dB_t$), but somehow I don't get how the above two equations are obtained!!
