equities - Is the stock price process a martingale or a Markov process?

Some people claim that the data-generating process for stocks is a "martingale" and that is has the "Markov property".

Are they unrelated? Is it that the Markov property implies some sort of martingale property, or is it the other way around?

How do you statistically test for such properties? How far from reality is it to assume such properties?

Answer

From what I remember, there is no real relation between Markov and Martingale, and my intuition was confirmed by this post.

Basically, it says that you can say neither of the following:

If A is Markov, then A is a martingale.

If A is a martingale, then A is Markov.

further down the post, you can find two counter examples:

$dX_t = a dt + \sigma dW_t$ is Markov but not a martingale

and

$dX_t = (\int_0^t X_s ds) dW_t$ is a Martingale but is not Markov.

As for the assumption of these properties being true, I think it really depends on how you see stock markets. My personal opinion being that no, the assumption is not very realistic.