I am going through Klenke's Probability Theory. Here are the definitions he presents for adapted

Definition 9.10 A stochastic process $X = (X_t,t \in I)$ is called adapted to the filtration $\mathbb{F}$ if $X_t$ is $\mathcal{F}_t$-measurable for all $t \in I$.

and for predictable

Definition 9.12 A stochastic process $X = (X_n,n\in \mathbb{N}_0)$ is called predictable with respect to the filtration $\mathbb{F} = (\mathcal{F}_n,n \in \mathbb{N}_0)$ if $X_0$ is constant and if, for every $n \in \mathbb{N}$ $X_n$ is $\mathcal{F}_{n-1}$-measurable.

The definition of predictable makes sense to me if $\mathbb{F} = \sigma(X)$, but when this is not the case, I do not know how to predict future values in $X_n$.

I constructed a concrete example. Let $(\Omega,\mathcal{F},\mathbb{P}) = ([0,1),\mathcal{B}(\Omega),\mu)$. Then, take the following process. Let $X_0(\omega) = 0$. Then, for $i \geq 1$, let $\omega_j$ be the jth digit in the decimal expansion of $\omega$ and $$X_i(\omega) = \begin{cases} 1 & \omega_j \text{ even} \\ 0 & \omega_j \text{ odd}.\end{cases}$$

If we let $\mathbb{F} = \mathcal{F}$ (giving out the original $\sigma$-algebra at each time step), then each $X_i$ is $\mathcal{F}_0$-measurable. Now we start the process and find that $X_0 =0$. How do I then predict the value of $X_1$?


The point you're missing is that the $\ t^\text{th}\ $ term $\ \mathcal{F}_t\ $ of the filtration is meant to represent all the knowledge available to you at the instant $\ t\ $, in the sense that, for every event $\ A\in \mathcal{F}_t\ $, either it is known that $\ A\ $ has already occurred, or it is known that $\ \Omega\setminus A\ $ has already occurred (i.e. it is known at that instant whether $\ \omega\in A\ $ or $\ \omega\in\Omega\setminus A\ $ ).

If $\ \left(X_n, n\in\mathbb{N}_0\right)\ $ is predictable with respect to a filtration $\ \left(\mathcal{F}_n, n\in\mathbb{N}_0\right)\ $, and $\ v=X_n(\omega)\ $ is the realised value of $\ X_n\ $, then at time $\ n-1\ $, even though you might not—and usually won't—know $\ \omega\ $, you do know that $\ \omega\in X_n^{-1}\left(\{v\}\right)\ $, because $\ X_n\ $ is $\ \mathcal{F}_{n-1}$-measurable, and so $\ X_n^{-1}\left(\{v\}\right)\in\mathcal{F}_{n-1}\ $, and hence you know that $\ X_n(\omega)= v\ $

In your example, by making $\ \mathcal{F}_0=\mathcal{F}=\mathcal{B}\left([0,1)\right)\ $ you are specifying that $\ \omega\ $ is already completely known at the start of the process, because all singleton events $\ \{\omega\}\ $ belong to $\ \mathcal{B}\left([0,1)\right)\ $.

  • $\begingroup$ Ok, I think I am understanding most of what you are saying. In the general case (your first paragraph), how is it that we know whether $\omega$ is in a given $\mathcal{F}_t$-measurable set or not? Also, just to double check, is there only one value of $\omega$ that is chosen at the start of the process and is plugged into every $X_n$? $\endgroup$ – sandaga Feb 5 '20 at 13:32
  • $\begingroup$ In the general case, how you're supposed to have acquired the knowledge represented by the filtration isn't specified. The filtration is simply the model of what is taken as being the information you acquire as time passes. $\endgroup$ – lonza leggiera Feb 5 '20 at 13:58
  • 1
    $\begingroup$ In practice (at least, in the limited contact I've had with it) the information is acquired by learning the values of some sets of random variables. In that case, if $\ \mathcal{S}_t\ $ is the set of random variables whose values you've learned by time $\ t\ $, then $\ \mathcal{F}_t\ $ would be the $\ \sigma$-algebra generated by $\ \left\{X^{-1}(B)\,|\,X\in\mathcal{S}_t, B\in\mathcal{B}\right\}\ $, where $\ \mathcal{B}\ $ is the Borel $\ \sigma$-algebra on the ranges of the $\ X$s in $\ \mathcal{S}_t\ $. $\endgroup$ – lonza leggiera Feb 5 '20 at 14:07

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