Is there any intuition behind the statement $E[X_\tau \mid \mathcal{F}_\sigma]=X_\sigma$ a.s. I mean I know that the interpretation of the conditional expectation and how to visualize it somehow but I kind of lose track when I see stopped processes as they are way abstract for me to make any real sense of it. I mean I can use the definitions and prove a few things but then I would really like to understand the relevance of these stopping theorems.
P.S: In my course on continuous time finance we did a lot of stopping theorems and their proofs which I think I understood but then I have no intuition Sorry if this question is not very precise but I have been struggling with this for quite some time and I felt compelled to ask
It is relatively intuitive. It says that your information of the past up to now doesn't help you to predict the future. Your expectation is neither higher is smaller of current's value. You mentioned studying finance. Those traders who follow trend for example , think it is not martingale and they follow the trend because they think the expected price in the future is not equal of today's value.