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Shannon entropy, $H(X) = -\sum_{i=1}^n p(x) \ln p(x)$ is a probabilistic measure of randomness or disorder within a random variable's probability distribution or histogram.

If we take rolling window segments, or snapshots, of a full-sample time series of stock returns $X$, can we expect the Shannon entropy of these windows to consistently increase or decrease as $t\rightarrow \infty$? i.e. each window has its own pdf.

or would their entropy merely be a function of the volatility (clustering) in each window, given that the spread of a distribution (volatility) and its randomness (entropy) are inextricably linked?

In thermodynamics, time entropy naturally grows with time, so mustn't there be a connection between the Shannon entropy and time entropy of a stock's returns?

How about the entropy of stock returns using expanding windows instead?

develarist
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    Just note, in thermodynamics, the $\Delta S \ge 0$, i.e. it grows or stay same. The entropy is same in case of reversible processes. I would expect something similar for stock. For example, for well established companies, $\Delta S \approx 0$. However, as uncertainty on market increases or decreases, also entropy would increases or decreases. Very nice question! – Martin Vesely Oct 19 '20 at 04:54
  • This can be of interest for you: https://quant.stackexchange.com/questions/879/can-the-concept-of-entropy-be-applied-to-financial-time-series/51817#51817 – Martin Vesely Oct 19 '20 at 04:57
  • Last comment to entropy in thermodynamics, it can also decrease but you have to do some work. The decrease in entropy has to be compensanted by its increase somewhere else. – Martin Vesely Oct 19 '20 at 05:01
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    This is closely related to whether stock returns are stationary. If they are, the unconditional distribution is not a function of time. Then the true entropy, defined using unconditional distribution should be constant. – fes Oct 19 '20 at 05:42
  • @MartinVesely what is $\Delta S$ in thermodynamics? – develarist Oct 19 '20 at 14:49
  • @fesman how does the formula for entropy change for the conditional, versus unconditional, distribution? – develarist Oct 19 '20 at 14:50
  • Instead of $p(x)$ you use $p(x | I)$ where $I$ is information available at say date $t$. I said unconditional distribution because stationarity does not rule out variation in conditional distribution that might induce variation in such conditional entropy. – fes Oct 19 '20 at 16:59
  • $I$ is information at $t$? how do I do that – develarist Oct 19 '20 at 17:55
  • It depends. You might e.g. have a view about the variance of the distribution based on some GARCH type model. Then after high recent volatility you increase the dispersion of your conditional distribution because variance is likely to be high for some time. – fes Oct 19 '20 at 18:34
  • @develarist: $\Delta S$ is change in entropy. – Martin Vesely Oct 20 '20 at 10:23

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