The lecture notes on stochastic processes that I am currently reading state that
The SDE given by $$dX_t = \mu X_t dt + \sigma X_t dW_t$$ (for known constants $\mu$ and $\sigma > 0$) is known as a geometric Brownian motion (GBM). The rate of change of $X_t$ is proportional to $X_t$ meaning that GBM never hits zero (or infinity by time inversion).
Could someone clarify for me what this means?
From Ito's lemma it follows that this GBM SDE has the following solution (see e.g. here https://en.wikipedia.org/wiki/Geometric_Brownian_motion)
$$X_t=X_0exp((\mu-\frac{\sigma^2}{2})t+\sigma W_t)$$
Now $X_t=0$ requires $W_t=-\infty$ and $X_t=\infty$ requires $W_t=\infty$. But the Wiener process takes values only on the reals $\mathbb{R}$. This follows from the normally distributed increments property of the Wiener process and the fact that the normal distribution is defined on the reals. Infinity is not a real number.