## 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 $$EX_T \neq EX_0$$.

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

• $$X$$ is not a MPP if and only if 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, which is usually taken to imply short-term prices should follow a martingale, but really implies they should follow some MPP.