Sure the unconditional expectation doesn't change, but that's kinda useless because it's the expectation given that you know nothing. The interesting part is studying what is next given what I know right now i.e conditional expectations.
And the martingale assumption i.e. "my best guest for tomorrow is the same as right now" is honestly a pretty sensible assumption for many things.
If I tell you $TSLA is at 200 right now, it's not unreasonable to assume it will be around 200 tomorrow.
If it's raining right now, it doesn't seem too far fetched to guess it will probably be raining in 1 minute.
etc.
And because you can prove so many things on martingales, it is often very very useful and powerful when you have something that isn't quite a martingale to think of a way to make it a martingale, prove whatever and then go back to the original object.
> If it's raining right now, it doesn't seem too far fetched to guess it will probably be raining in 1 minute.
That's a bit of an unfair example, though. If the Tesla stock is at 200 right now, the martingale property implies that I should expect it to be at 200 not just next minute or tomorrow, but also next week, two years from now, next decade, and so on. A martingale is not restricted in its time scale.
(This is using clearly expectation in the technical sense. The stock price may well go up, or go down, but we can't tell which or how much, so in the grand scheme of things, we're better off assuming it won't move at all.)
To expect a value of 200 means to have the average of 200 from this point in time onwards, assuming stock price is random walk. Not that the value tomorrow will be exactly 200. It could be 200, 201, 199, 202, 198 etc. the average expected is 200. If you possess no external knowledge such as insider information, then random walk is a sensible and obvious choice for stock price.
yeah there's a built in assumption that the behavior of the function we're estimating with martigale is continuous near the limit of the guess, and thus predictable over the interval of the guess and the prediction.
Sure the unconditional expectation doesn't change, but that's kinda useless because it's the expectation given that you know nothing. The interesting part is studying what is next given what I know right now i.e conditional expectations. And the martingale assumption i.e. "my best guest for tomorrow is the same as right now" is honestly a pretty sensible assumption for many things.
If I tell you $TSLA is at 200 right now, it's not unreasonable to assume it will be around 200 tomorrow.
If it's raining right now, it doesn't seem too far fetched to guess it will probably be raining in 1 minute.
etc.
And because you can prove so many things on martingales, it is often very very useful and powerful when you have something that isn't quite a martingale to think of a way to make it a martingale, prove whatever and then go back to the original object.