Let us have a toss of two coins and our $\Omega=(hh,tt,ht,th)$. 1) give a filtration so that all stochastical processes are also adaptable and also predictable. 2)given the filtration $F_{0}= trivial,F_1=(F_0,(ht,hh),(th,tt))$ and $F_2$= power set: give an example of a process which is not adaptable and one which is adaptable but not predictable.
I understand that in the natural filtration every process is adaptable and in the filtration of complete information every process is predictable but I don't know something where both of them are the same.
not adaptable maybe would be a process where $X_0$ is for example not constant but I have no idea of something which might be adaptable but not predictable.
It's the very first time that we are getting to know this concepts and in the lecture they were explained very briefly so I still don't have a feeling regarding this so any simple examples/hints would be much appreciated.
Thanks in advance.
To be adapted, a process $X$ must satisfy: (0) $X_0$ is constant, (1) $X_1$ depends only on the result of the first coin toss.
An adapted process is predictable if: 0) $X_0$ is constant, (1) $X_1$ is constant, (2) $X_2$ depends only on the result of the first coin toss.
Consider the process defined by $X_0=0$; $X_1$ is the indicator of the event that the first coin toss results in Heads; $X_2$ is the event that the second coin toss results in Tails. Is $X=(X_0,X_1,X_2)$ adapted? Is $X$ predictable?