Time Reversal of Brownian Motion and Mean-Reverting Processes
Summary
The document asks whether reversing the order of observations or inverting a time series preserves Brownian motion, and whether mean reversion would become momentum under reversal. The responses distinguish a driftless Brownian motion from processes with drift: they state that reversing a driftless Brownian path still gives Brownian motion, while an upward drift changes direction when time is reversed.
The discussion also cautions that a mean-reverting series is not Brownian motion and that there is no general rule turning mean reversion into momentum in real market data. One answer describes reversal of a mean-reverting process as making it less stable in the opposite direction. These are brief conceptual replies rather than a full treatment of definitions, time horizons, or conditions for time reversal. The suggested use in judging a hedge is speculative; the document offers no empirical test or evidence that a reversed series reliably diagnoses hedge quality.
Key ideas
- A driftless Brownian motion remains Brownian under time reversal, according to the responses.
- Time reversal changes the direction of a process with drift.
- Mean-reverting processes are distinct from Brownian motion.
- There is no universal rule that turns mean reversion into momentum when observations are reversed.
- The proposed use of reversal to assess a hedge is speculative and unsupported by evidence in the document.
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Full text
# Reverse chronological time series / inverse time series # Reverse chronological time series / inverse time series 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? ## Answer by user3264325 (score 1) https://quant.stackexchange.com/a/39889 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. ## Answer by Ezy (score 1) https://quant.stackexchange.com/a/42194 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. ## Answer by Sebapi (score 0) https://quant.stackexchange.com/a/43074 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.
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