$X_t$ follows the Ito process as described by the following stochastic differential equation $$dX_t=b(X_t)dt+dB_t\quad , \quad X_0=x$$
and $b(X)$ satisfies Lipschitz Coditions.I want to show for every $M>0$ and $t\in[0,\infty)$, we have $$\mathbb P(X_t\geq M)>0$$
Any help? Any direction?
I'm working with Oksendal "Stochastic Differential Equations# (mostly) and also have Shreve "Stochastic Calculus" and found nothing similar to work with.