I came across the following statement in Sebastian Bossu's book "Advanced Equity Derivatives", page 27. He says that the time-homogeneous diffusion process $dX_t=a(X_t)dt+b(X_t)dW_t$ (coefficients $a$, $b$ are functions of $X_t$) will be bounded by $[l,u]$ if $a(l)=a(u)=b(l)=b(u)=0$.
Is there any rigourous proof of this fact? Because for example if we take $dX_t=X_t^2dW_t$ (I took $l=0$, $u=\infty$). Then applying Ito Lemma we obtain $d\frac{1}{X_t}=-dW_t+X_tdt$. If $X_t$ was bounded by 0, so $Y_t=\frac{1}{X_t}$ must also be positive. I don't think this is true :)