even if you could, you would be better off working for a bank, rather than risking your own money.

That depends on the particulars. In any case, this is more the bailiwick of hedge funds rather than banks nowadays.

suppose I had discovered an entirely novel technique for predicting the market

You would have to demonstrate that it works and convince people with money to give some of that money to you to trade.

You can try to persuade people with just your technique (and backtesting results) under an NDA, but without a track record it will be hard -- or the deals offered to you won't be good. You also can trade your own money -- in whatever amounts you have -- for a while in a separate audited account. A good track record and a demonstrated willingness to risk your own money on your idea will help you persuade other people that the idea works.

Remember, you don't have to mathematically prove anything, all you have to do is convince people with money to give you some :-)

how easy it is for AI to acquire massive resources by dominating the world's financial markets.

For a fully developed AI it will be easy, but I am not sure why would it bother. And if you're thinking about trading "AIs" developed on Wall St. and such, these are very likely to be narrow tool-like AIs with very specific and limited goals.