Example of filtration in probability theory

Question

I'm studying Martingales and before them filtrations. Given a probability space $(\Omega, F, P)$ I define a filter $(F_n)$ as a increasing sequence of $\sigma$-algebras of $F$, such that $F_t \subset F$ and $t_1 \leq t_2 \Longrightarrow F_{t_1} \subset F_{t_2}$. Here comes my question: How can the $F_t$'s be $\sigma$-algebras and subsets of $F$ without being exactly equal to $F$? I suppose that $F_t$'s being $\sigma$-algebras mean that they are $\sigma$-algebras with respect to the measure space $(\Omega, F)$. Can anyone explain why they are not necessarily equal to $F$ and give an example where this is obviously false?

Answer 1

Take the following simple model: a stochastic process $X$ that starts at some value $0$. From that value, it can jump at time $1$ to either the value $a$, either the different value $b$. And at time $2$, it can jump to $c$ or $d$ if it was in $a$ at time $1$, it can jump to $e$ or $f$ if it was in $b$ at time $1$. In other words, there are four possible paths for the variable $X$: $\omega_1=0\to a\to c$, $\omega_2=0\to a\to d$, $\omega_3=0\to b\to e$ and $\omega_4=0\to b\to f$. They constitute our space of outcomes

$$\Omega=\{\omega_1,\omega_2,\omega_3,\omega_4\} \; .$$

Hence, $\Omega$ is the space of possible paths for $X$. On the other hand, you can define a filtration as follows

$$\begin{eqnarray} \mathcal{F}_t = &\{\emptyset,\Omega\} &, 0\leq t <1 ; \\ \mathcal{F}_t = &\{\emptyset,\{\omega_1,\omega_2\},\{\omega_3,\omega_4\},\Omega\} &, 1\leq t <2 ; \\ \mathcal{F}_t = & \mathcal{P}(\Omega) &, 2\leq t .\end{eqnarray}$$

with $\mathcal{P}(\Omega)$ the power set of $\Omega$. In this sense, the filtration is a reflection of the info you have at any time. In the beginning, you don't know which path the stochastic variable will follow, so your filter does not contain more than the events $\emptyset$ and $\Omega$, but in the next step, you can arrive at one of the values $a$ or $b$. Therefore you have two extra events in your set you can speak about. And finally at the final time, you have access to all possible events. As you can see $\mathcal{F}_1 \subset \mathcal{F}_2$ but $\mathcal{F}_1 \neq \mathcal{F}_2$.

Answer 2

Another simple example. The natural filtration when we model tossing a die twice in a row. Here is how it works. Let $X_1$ be the outcome of the first toss. So the values of $X_1$ are in the set $\{1,2,3,4,5,6\}$. Let $X_2$ be the outcome of the second toss.

As a sample space, we can take $\Omega = \{1,2,3,4,5,6\} \times \{1,2,3,4,5,6\}$, the set of all ordered pairs chosen from the set $\{1,2,3,4,5,6\}$. If $\omega \in \Omega$, then $\omega$ is an ordered pair, say $\omega = (\omega_1,\omega_2)$. Let the two random variables be $X_1(\omega) = \omega_1$ and $X_2(\omega) = \omega_2$. An "event" is a subset of $\Omega$.

[Note: a true probabilist thinks the first paragraph is quite natural, and the second paragraph is very artificial.]

The "times" that are relevant are: time $0$, before any tosses have been done, time $1$ after the first toss but before the second toss, and time $2$, after the second toss. For each time $t$, the sigma-algebra $\mathcal F_t$ is "the information known at time $t$". We have $\mathcal F_0 \subset \mathcal F_1 \subset \mathcal F_2$, with strict inclusion in all cases.

Now let's work out what these are.

$$ \mathcal F_0 = \{\varnothing, \Omega\} $$ since at time $0$ we have no information about which sample point is the true state of affairs. There are only two events such that we know whether it occurs: event $\Omega$ definitely occurs, and $\varnothing$ definitely does not occur. For any other event, we do not know whether or not it occurs. $$ \mathcal F_1 $$ consists of $2^6$ events: all sets of the form $A \times \{1,2,3,4,5,6\}$, where $A \subseteq \{1,2,3,4,5,6\}$. This is because, at time $1$, we know what the outcome $X_1$ was, that is, we know the first coordinate of $\omega$, but we do not know the second coordinate of $\omega$. The events in $\mathcal F_1$ are events that contain no information about the second toss $X_2$. Given an event $U$ in $\mathcal F_1$, at time $1$ we definitely know whether or not $U$ occurs. But given an event $V$ not in $\mathcal F_1$, at time $1$ we possibly do not know whether $V$ occurs. $$ \mathcal F_2 $$ consists of $2^{36}$ events: all subsets of $\Omega$. This is because, at time $2$ we know exactly what $\omega$ is, so, for any of the $2^{36}$ events $U$, we know whether or not $U$ has occurred.

[Probability language $$ \text{event $U$ occurs} $$ translates to set theory language $$ \omega \in U. $$ As you work with this, you will get more experience translating back and forth between them.]