Sample log-returns at your any frequency (hourly, daily, weekly, monthly etc), get the SD then annualise to get the vol in the usual way. If we find that the annualised vol is the same whatever the sample frequency, I think it follows that the underlying process is GBM. But how do we prove that?
I believe there is a proof in Shreve vol II but lockdown prevents me getting to the library. Is there a proof online anywhere? Thanks
[EDIT] Intuitively (1) the returns must be independent, otherwise it is easy to show that the required condition (annualised vol is the same, whatever the sampling frequency) fails. And (2) the returns must themselves be normally distributed, by central limit. But properties (1) and (2) define GBM.