So for stochastic process $X_k$, We can define probability space, and filtration $\mathcal{F}_k$.
As far as I know, as $\mathcal{F}$ is sigma algebra, filtration represents sequences of events that can be assigned probability at some time period. Is this correct?
And if so, this means that as time flows, the number of sequences of events that can be assigned probability increase. Is this correct?
If I am mistaken, please correct the errors.
Best Answer
I don't think you want to interpret the filtration $\mathcal{F}_t$ as the events that can be assigned a probability at the time $t$. It is more natural to interpret it as the information present at time $t$. Let me explain:
The usual setup is that we have a probability space $(\Omega,\mathcal{F},P)$, i.e. $\Omega$ is a non-empty set, $\mathcal{F}$ is a sigma algegra on $\Omega$ and $P$ is a probability measure on $\mathcal{F}$. Here we have already said that the sets/events in $\mathcal{F}$ are the sets/events we can assign a probability, and so this does not vary over time.
Now we can equip our probability space with a filtration $(\mathcal{F}_t)_{t\geq 0}$, that is $\mathcal{F}_t$ are sigma algebras with $\mathcal{F}_t\subseteq \mathcal{F}$ for all $t$ such that $$ \mathcal{F}_s\subseteq \mathcal{F}_t\quad\text{whenever }s\leq t.\tag{1} $$ So you can see why we don't want to think of $\mathcal{F}_t$ as the sets/events that we can assign a probability because this is at any time given by the whole of $\mathcal{F}$.
Instead, think of $\mathcal{F}_t$ as the information present at time $t$. Then $(1)$ tells us that we are not getting dumber with time which is a reasonable assumption. If $X:\Omega\to\mathbb{R}$ is a random variable that is $\mathcal{F}_t$-measurable, then it means that we can determine $X$ based on the information available at time $t$ because $$ X^{-1}(B)=\{X\in B\}\in\mathcal{F}_t,\quad \text{for all Borel sets }B. $$