Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes.

What is known about their mathematical properties ?

I know there is huge (too huge) literature (see e.g. MO-Financial Mathematics Books) around it, I am familiar with some ideas, like below, but would be grateful for any comments/suggestions.

1) The individual distributions are better modelled by heavy-tailed distributions, rather than by normal distribution, reflecting that sometimes prices change heavily in short time periods. (See e.g. MO: Is there any straightforward way to substitute for Gaussian/Brownian assumptions in financial mathematics?).

2) To some extent they are similar to Brownian (log-Brownian) motion, more precisely the increments are independent at least at some time scales (which means that you cannot win money), (however there are other claims that short time increments are correlated).

3) There are claims that fractal (Hausdorff ) dimension is near to 1.5 (the same as for Brownian motion).