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If a time series follows a Brownian motion (BM), is it true that the inverse ts and reverse chronological ts is also a BM?
What if the ts exhibits mean reversion tendencies? Would these tendencies become a momentum tendencies?
Is this a useful/common field of studies in the quant world?
There’s a definition of equilibrium in physics which says that if you c cannot tell whether a time series is going forward or backwards then you are in a state of equilibrium.
A possible application would be seeing whether you can spot a hedged portfolio’s time series has been flipped backward or forward.
If it’s not obvious you prob have a good hedge.
the time reversed brownian-motion is still a brownian motion.
see for instance this (example 15.5)
https://www.stat.berkeley.edu/~pitman/s205s03/lecture15.pdf
Now if your time series displays mean-reversion it is not BM.
I am not sure about what exactly you mean by a mean-reverting time series to display momentum but I am pretty certain the correct answer is no in the sense you have no general answer to this type of question in actual market time series.
Certainly in the quant world people are interested by knowing if time series exhibit mean-reversion or trending tendencies.
A time reversed BM with no drift has all the BM properties.
Drift up will become drift down.
A mean-reverting process has vanishing variance in one direction, but with time reversed, this becomes mean-revulsion, which makes the process less stable in the opposite direction.
