I am working on high frequency data and try to implement an Ornstein-Uhlenbeck mean reverting process to model short-term momentum \(\alpha\).
$$ {d\alpha_t=\eta(\theta-\alpha_t)dt+\sigma dB_t} $$
The timestamp interval between each best bid-ask sample is 0 - 30 milliseconds, irregularly. I am confused on how to sample \(\alpha\). If I define it as the difference of mid price between each timestamp, then \(\eta\) tends to be infinity( estimated based onJose Carlos Garca Franco's paper Maximum likelihood estimation of mean reverting processes). I guess the reason is that the sampling frequency of the raw data is too high. There is some cases that the mid price changes in less than 1 millisecond, but the prceision of timestamp is up to 1 millisecond.
If I resample the data at frequency 30, 50, 100, 500 milliseconds, I can get meaningful parameters, but they vary a lot. Does it mean momentum reverse at various rate at various time scale? If resampling data is ncessary, then how to choose the resampling frequency?