Hey,
I have an understanding issue. Why is the E[X(t)|F(t)] = X(t) true? All I see here is that given a adapted-filtration space X then the expected value of X is still a random variable X. The only reason this makes sense to me perhaps would be through the tower of expectations i.e. E[E[X(t)|F(t)]] = E[X(t)]. If I try to dive deeper into how filtrations work I get confused. My understanding of the tower of expectations goes as follows: If E[X] = E_X[E_Y[X|Y]] but in this scenario why would the E_F(t)[X(t)|F(t)] = X(t)?
Last edited by a moderator: Aug 12, 2019