## Martingale, for practical purposes

In several areas of mathematics, such as computational complexity and cryptography, there is a
meaningful and important distinction between "there exists ....." and
"it is practical to find ........".
This distinction is not currently made within mathematical probability,
but here is a setting where it would be interesting to do so.
Consider a stochastic process
\(X = (X_t, 0 \le t \le 1)\).
A familiar theorem states (in contrapositive form,
and ignoring technical conditions)

\(X\) is not a martingale if and only if **there exists** a stopping time \(T\) such that
\(\mathbb{E}X_T \neq \mathbb{E}X_0\).

I would like to define a class of processes, say "martingale for practical purposes" (MPP)
by
\(X\) is not a MPP if and only if **it is practical to find** a stopping time \(T\) such that
\(\mathbb{E}X_T \neq \mathbb{E}X_0\).

**Problem**. Make sense of this definition.

This arises from the *efficient market hypothesis*, which is usually taken to imply short-term
prices should follow a martingale, but really implies they should follow some MPP.

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