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For a real valued random variable (or probability distribution), the (relative) entropy is used to quantify how random it is. Provided a stochastic process, how can we determine whether it is ''very random'' or ''just a little random''?

More precisely, we consider the boxing match between players A and B. The match ends if either of the two conditions below is satisfied :

  1. One player KO (knockouts) the other and the match is interrupted immediately;
  2. No KO occurs before time $T$ and the match ends at $T$.

If A wins, then $1$ point is gained; otherwise $1$ point is lost. Denote by $X_t$ the expected point that A may gain at time $t$. Then $X_T\in \{1,-1 \}$. If we assume further A is as same good as B, then $X_0=0$ and $(X_t)_{0\le t\le T}$ is a martingale. Which is the most exciting match? Or namely, which is the most random martingale starting at zero and ending at $1$ or $-1$?

Any answers, comments (on the rigorous mathematical formulation) or references (related to the randomness of martingales) are highly appreciated!

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    $\begingroup$ I assume that your martingale is continuous (if not let me know). It is known that every continuous martingale starting from zero is a time-changed BM, i.e. $X_t=W_{\langle X\rangle _t}$ with $W$ being a BM (w.r.t. some suitable filtration). Thus we may reformulate your question as follows : Find the time-changed BM as "random" as possible. In particular with $\tau:=\langle X\rangle _T$, we see the BM stopped at $\tau$ has the distribution $1/2\delta_{-1}+1/2\delta_{1}$. Thus we can deduce that $\tau:=\inf\{t\ge 0: |W_t|=1\}$. So you may refer to the literature related to Skorokhod embedding $\endgroup$
    – user128095
    Oct 21, 2021 at 17:02
  • $\begingroup$ While for the randomness of martingales, this is beyond my knowledge $\endgroup$
    – user128095
    Oct 21, 2021 at 17:03
  • $\begingroup$ If this is a discrete time process, do you not measure randomness by Kolmogorov complexity or the like? Beyond that, there is a theory of algorithmic randomness that belongs more to logic than probability theory, discussed in the book "Algorithmic Randomness and Complexity" by Downey and Hirschfeldt. Note that "most exciting" is probably not the same as "most random", since the game is exciting if it is slightly guessable. $\endgroup$
    – none
    Oct 23, 2021 at 7:44
  • $\begingroup$ @none Thanks very much for letting me know about Kolmogorov's complexity. I find the book at "hal-lirmm.ccsd.cnrs.fr/lirmm-01803620/file/…", while I do not see why the algorithmic randomness can be related to the randomness of martingales. Could you please specify a bit more on this issue? Thank you very much! $\endgroup$
    – user420828
    Nov 3, 2021 at 16:50
  • $\begingroup$ Cross-posted: mathoverflow.net/q/406733/37212, cs.stackexchange.com/q/145981/755. Please do not post the same question on multiple sites. Please pick one site where you want the question to appear and let us know what that site is; and update the question on that site based on the feedback you have received on both sites. Thank you! $\endgroup$
    – D.W.
    Nov 22, 2021 at 9:31

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