Drift irrelevance on high frequency data

Question

Let's assume that price of a certain asset follows Brownian Semimartingale process with a drift term and a Brownian-driven continuous part (no jumps for simplicity). In literature it is often stated that on high frequency data (i.e. order of seconds or even ticks) drift becomes irrelevant both economically and statistically (so we can safely assume that making it equal to zero will have close to none impact on our analysis). While I understand the notion behind it, I'd like to have some kind of formal proof of this statement, or at least some solid empirical evidence.

Answer 1

This is no means a rigorous proof but I believe it may help.

Let's suppose you have Brownian motion under the normal conditions.

$dX_t = \mu X_tdt + \sigma X_tdW_t$

The drift $\mu$ and volatility $\sigma$ are constants for each time step. Now let's suppose we 'zoom in' to a smaller time step $s = t/2$. Our drift portion of the equation goes to $\frac{\mu}{2}$ while our volatility goes to $\frac{\sigma}{\sqrt{2}}$

If we continue this pattern we will quickly see that our drift portion of the GBM equation decreases much more quickly than our volatility portion.

So in one year there are 3153600 seconds. If we were to calculate our drift and volatility based on one year and then use seconds as our time step our mean would be $\frac{1}{3153600}$ as large while our volatility would be $\frac{1}{1775.83}$ as large.