Does this SDE have a unique solution: $$dX_t=X_tB_tdB_t+X_tB_tdt, \quad X_0=1?$$
I have to check if the Lipschitz condition holds and if the growth condition holds.
I presume not, as the Brownian motion can take any value (it is unbounded, so the growth and the Lipschitz condition are not satisfied. But how can I formally prove this, if my intuition is indeed correct?
How can one generally prove/disprove the conditions for existence and uniqueness, besides showing if it is differentiable?