Questions regarding filtration - more information

Question

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.

Answer 1

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 $$ X^{-1}(B)=\{X\in B\}\in\mathcal{F}_t,\quad \text{for all Borel sets }B. $$

Answer 2

A concept of $\sigma$-algebra is beneficial in two aspects. Firstly, it describes sets with a nice structure to define nice measure over, like Borel $\sigma$-algebras.

A second aspect is more relevant to your questions, and concerns the information carried by the $\sigma$-algebra. The main hint is given in the following theorem:

Let $X:(\Omega,\mathcal F)\to (A,\mathcal A)$ and $(Y,\mathcal F):\Omega\to (B,\mathcal B)$ be two measurable maps. If $Y$ is $X$-measurable, then there exists a measurable map $$ g:(A,\mathcal A)\to(B,\mathcal B) $$ such that $Y = g\circ X$.

The point is that any $\sigma$-algebra carries the whole information about the "level sets" of the functions measurable w.r.t. such $\sigma$-algebra and thus it can represent an information.

The probability in fact has little to do with filtrations. Given any measurable space $(\Omega,\mathcal F)$ the filtration is an ordered collection of sub-$\sigma$-algebras $(\mathcal F_t)_{t\in T}$ such that $s\leq t$ implies $\mathcal F_s\subseteq \mathcal F_t\subseteq \mathcal F$. In case $T = \Bbb R$, the $\sigma$-algebra $\mathcal F_t$ contains an information of what happened up to the time moment $t$. As an example, if you consider an adapted stochastic process $X_t$, i.e. $$ X_t:\Omega\to \Bbb R,\quad X_t \text{ is }\mathcal F_t\text{-meas.} $$ then $\mathcal F_t$ contains an information about all value of $X_s$ for $s\leq t$.