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.
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Re: GBM
There is a related theorem called "Levy's characterization of Brownian motion", with lots of sources online. It requires two assumptions: (i) same as yours on the variance and (ii) that the process is a continuous martingale. Whether or not it can be extended to "drifting Brownian motion" by dropping the "martingale", or dropping (ii) altogether, I don't know.
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Re: GBM
Thanks Alan!There is a related theorem called "Levy's characterization of Brownian motion", with lots of sources online. It requires two assumptions: (i) same as yours on the variance and (ii) that the process is a continuous martingale. Whether or not it can be extended to "drifting Brownian motion" by dropping the "martingale", or dropping (ii) altogether, I don't know.
Re: GBM
This isn’t exactly what you want, but your reference to the central limit theorem suggests that you may want to take a look at Donsker’s theorem, which is the equivalent convergence result for paths in a suitable function space. Evidently Donsker didn’t get it quite right, but it has been subsequently fixed up.
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