I am currently reading the book 'High Frequency Financial Econometrics' by Jacod and Ait-Sahalia and have a problem with the following statement in chapter 2:
'The usual rationale for first-differencing a time series is to ensure that when T [the time horizon] grows the process [X, used for modeling the log of an asset] does not explode: in many models, the process X may be close to a local martingale and thus the discrete date $X_{i\Delta}$ may be close to exhibiting a unit root, whereas the first differences of X will be stationary or at least non-explosive.'
$\Delta$ is the frequency rate at which the process X is sampled with mesh tending to zero.
I am not familiar with the notion of unit root. What do the authors mean by it? Is someone able to give an example, when the process X could explodes? I am also not sure, waht it means to explode.
Thank you in advance!