Why does the market need to be beatable if it's inefficient?
If the market is chaotic, you can't beat it just because you're smart.
If the market is chaotic, you can't beat it just because you're smart.
1. Just buy when the price is low and random variation will move the price higher eventually (or short when the price is high).
2. Ignore the market except to buy when a company is underpriced, then just have the company declare dividends or whatever and directly eat the profit that the market is stipulated to have underpriced.
A very small investor in a random market might have difficulty with either of these strategies, but a reasonably well-capitalized investor would not.
If you want to think up some kind of complicated model of difficult-to-take-advantage-of company pricing that is hard to exploit, really ask yourself whether that model is grounded in anything other than, "I'm mad about the EMH."
1. How do you know when the price is high or low? The market can remain irrational for a very long time (longer than you can stay solvent).
2. You can easily lose money shorting when you're right. People do it all the time.
You can't blow up.
You're not the House at the casino.
This is what I mean when I say that it's usually not helpful to try to make claims about the world based on, "I'm really mad about the EMH." You end up making silly claims.
Geometric Brownian motion can't take negative values either, so I don't know what you think this proves. While it's true that stock prices don't literally follow geometric Brownian motions (+drift), you can't tell the difference from stock graphs. (You can tell the difference with statistical tests if you know what to look for, e.g. volatility clustering, but humans can't tell by eye, and naive classification ML models will not perform well.)
I will note that "stocks are priced via geometric browning motion" would satisfy the EMH. And that the underlying growth trend of GBM, in a real world, is, like, "the stock market actually values real things about the company that you can find out, but which other people have already found out."