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I know that, given $\mathbb{I}\subset[0,\infty]$, we call a stochastic process $(X_t)_{t\in\mathbb{I}}$ adapted to the filtration $\left(\mathcal{F}_t\right)_{t\in\mathbb{I}}$ if $X_t$ is for each $t\in\mathbb{I}$ an $\mathcal{F}_t$-measurable random variable.

At this point, I cannot figure out - even intuitively - an example of a stochastic process which is not adapted. If for example one considers natural times, with $t=2$, how could it be possibile that the value of the r.v. $X$ at time $2$, that is $X_2$, is not $\mathcal{F}_2$-measurable (that is, not measurable wrt the information set available up to time $2$)?

Could you please give me some counterexample?

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Consider a real-valued Brownian motion $(B_t)_{t \geq 0}$ in its natural filtration, $\left(\mathcal{F}_t\right)_{t \geq0}$. Then, if we define a new process $(X_t)_{t \geq 0} = (B_{t+1})_{t \geq 0}$, then this is not adapted to the specified filtration. Indeed, for any $t > 0$, $X_t$ is only $\mathcal{F}_{t+1}$ measurable, but not $\mathcal{F}_t$ measurable by definition of the Brownian motion.

The intuition behind this is that we have constructed a process with a time lag, so that although we are in the right filtration, we technically only started the new process at time $1$, when we play the stopwatch and we start indexing time from $0$. So really when we try to observe the process $X$, we need to look ahead in the future to determine its value, as opposed to the present being sufficient.

Another example is to have two independent processes and consider the filtration generated by one of them. Then, the other process is not adapted with respect to this filtration. The reason is that knowledge of the first process tells us nothing about the second, so merely observing the first cannot lead to determining the second. This is a bit weird, but the key is to remember that filtrations are actually much more restrictive than we like to think. A filtration doesn't necessarily tells us everything that's happened in the universe up until a certain time, but rather focuses on one particular experiment or simulation (or multiple, by considering the filtration generated by the $\pi$-system of the union of several such filtrations). If two people start throwing dice on opposite ends of the world at the very same time, this corresponds to two different filtrations and each of the processes is not adapted with respect to the filtration generated by the throws of the other person (unless they're on the phone).

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  • $\begingroup$ I really like the accuracy of your answer as well as the ability to make things easy by means of examples. Have a nice day. Very compliments @spetrevski $\endgroup$ Jun 27, 2020 at 11:06
  • $\begingroup$ Thanks! Glad that this was useful :) $\endgroup$
    – spetrevski
    Jun 27, 2020 at 11:32

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