Sorry, first question here, so I apologize if the formatting is not great. I have a Stratanovich SDE given by
$$d\theta_{1}=\sin(\theta_{2})\circ dW_{1}-\cos(\theta_{2})\circ dW_{2}$$ $$d\theta_{2}=\frac{\cos(\theta_{1})}{\sin(\theta_{1})}\cos(\theta_{2})\circ dW_{1}+\frac{\cos(\theta_{1})}{\sin(\theta_{1})}\sin(\theta_{2})\circ dW_{2}+dW_{3}.$$
I would like to convert it to an Ito SDE and following the steps in Evans, "An introduction to Stochastic Differential Equations" (page 123) I found $$d\theta_{1}=\sin(\theta_{2})dW_{1}-\cos(\theta_{2})dW_{2}$$ $$d\theta_{2}=-\frac{\cos(\theta_{1})}{2\sin^3(\theta_{1})}dt +\frac{\cos(\theta_{1})}{\sin(\theta_{1})}\cos(\theta_{2}) dW_{1}+\frac{\cos(\theta_{1})}{\sin(\theta_{1})}\sin(\theta_{2}) dW_{2}+dW_{3}.$$ In other words the drift term for $d\theta_{1}$ is zero both in the Ito and Stratanovich representations and an additional drift term appears only in $d\theta_{2}$. In Infinitesimal generator of the Brownian motion on a sphere the accepted answer has, towards the end, a virtually identical conversion, giving instead
$$d\theta_{1}=\frac{\cos(\theta_{1})}{2\sin(\theta_{1})}dt+\sin(\theta_{2})dW_{1}-\cos(\theta_{2})dW_{2}$$ $$d\theta_{2}=\frac{\cos(\theta_{1})}{\sin(\theta_{1})}\cos(\theta_{2}) dW_{1}+\frac{\cos(\theta_{1})}{\sin(\theta_{1})}\sin(\theta_{2}) dW_{2}+dW_{3}.$$
Thus, it is the $d\theta_{1}$ term that gets the additional drift and $d\theta_{2}$ instead has identical drift in both representations. This last result seems correct because I verified that half the Laplacian is the infinitesimal generator of the process. So, what is that I got wrong?
