Yes thank you very much ! I achieved the proof by myself yesterday, and I used the comparison with the deterministic situation as well. I have previously solved the deterministic equation in all possible case so the fact is implied by this previous work. Your second proof is quite clear though :)

Great, thanks for the trick ! But if I'm not missing something, I think that we have just showed that $X_t$ diverges almost surely to infinity. We do not have showed that it's in finite time. But I should be able to conclude by myself though.