Sure we don't know the distribution. But it's a meta problem. We can know without knowing it all. There are shades of gray btw Perfect Information & Maximum Uncertainty.
I agree with K-Pia, more than Sergio. Stock market is NOT perfectly random. There is some tendency, however NOT MUCH. For explanation, I would say that 90% random and 10% causality. There will be lots of application of statistics, to stock market, specially in trading for profit. I don't care if someone such as Sergio does NOT use any statistics knowledge for trading.
Quite close to Brownian motion (Normal distribution) and also should see the Log-Normal since the market behaves under compounding process. Very very few are under compounding scheme in the world and most other area behaves under Euclidean ("simple" instead of "compounding"). Many reference books also show this fact.
The problem is not knowing the distribution; it is fat tails and black swans that kill profit and cause ruin. The probability of such events is much higher than that predicted by normal.