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)

$X$ is not a martingale iff there exists a stopping time $T$ such that $EX_T \neq EX_0$.

I would like to define a class of processes, say "martingale for practical purposes" (MPP) by

$X$ is not a MPP iff it is practical to find a stopping time $T$ such that $EX_T \neq EX_0$.

Problem. Make sense of this definition.

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