Consider an SDE \begin{equation} dX_t=f(X_t,t)dt+b(X_t,t)dW_t \end{equation} Suppose firstly that the coefficient are Lipschitz continuous. So by the theorem of existence and unicity I have that exist a strong solution and this is unique. So, can I say, that fixed an $\omega$ two trajectory can't intersect each other?
If the coefficient of the diffusion term isn't Lipschitz,(like $\sqrt{...})$, and a unique solution exist can be that the two trajectories intersect?