Is it correct to say that any martingale (e.g., a GBM or anything similar), taking values in some real state space and not converging to a degenerate random variable, has the property that, given a value $y$ in that space, the probability of going above it once (and, therefore, infinitely many times), no matter where we start from, is $1$ as time diverges?
Can I correctly say the same holds for semimartingales, and, in general, what is the most general family of processes that, if I start from any value, I will always be able to both exceed any value $y_2$ and also go below any value $y_1$ ($y_1 < y_2$) of its state space, at least once, with a probability of $1$, as time goes to infinity?
What is a formal justification for these facts?
This holds for martingales with $\langle M \rangle_\infty = \infty$. Suppose that $M$ is a such a martingale, then by the Dambis-Dubins-Schwarz theorem, $M_t$ is a time-changed Brownian motion. In particular,
$$P\left(\sup_{0 \leq s \leq t} M_s \geq a \right) = P\left(\sup_{0 \leq s \leq t} B_{\left\langle M\right\rangle_s} \geq a \right) \to 1$$ as $t \to \infty$, since Brownian motion is recurrent.