For a stopping time T, define $\mathcal{F}_T$ by
$\mathcal{F}_T={A \in \mathcal{F}:A \cap \{T \le t\} \in \mathcal{F}_t, \text{for each t.}}$
Verify that $\mathcal{F}_T$ is a $\sigma$-algebra.
To prove this I know we must show that $\mathcal{F}_T$ satisfies the three properties of $\sigma$-algebras, ie that
A $\sigma$-algebra $\mathcal{F}$ of subsets of a set X is a collection $\mathcal{F}$ of subsets satisfying the following properties;
(1) $\emptyset$ $\in$ $\mathcal{F}$
(2) if A $\in$ $\mathcal{F}$ then it's compliment $A^{c}$ is also in $\mathcal{F}$
(3) if $A_1,A_2,…$ is a countable collection of sets in $\mathcal{F}$ then their union $\cup^{\infty}_{n=1}A_n \in \mathcal{F}$
I have attempted this myself but am concerned I am not being precise enough. Any hints or comments are gladly welcome.
For (3):
Suppose $A_1,A_2,…$ is a countable collection of sets in $\mathcal{F}_T$
Then $A_n \in \mathcal{F}$ : $A_n \cap \{T \le t\} \in \mathcal{F}_t$ for all n by definition
Then $\cup^{\infty}_{n=1}(A_n \cap \{T \le t\})\in \mathcal{F}_t$
Then $(\cup^{\infty}_{n=1}A_n) \cap (\{T \le t\})\in \mathcal{F}_t$
Then $(\cup^{\infty}_{n=1}A_n) \in \mathcal{F}_T$
For(2) I am less convinced
My reasoning would be that if $A \in \mathcal{F}_T$ then $A \in \mathcal{F}: A \cap \{T \le t\} \in \mathcal{F}_t$
We know that $A \cap \{T \le t\} \in \mathcal{F}_t$
Now $A^c \in \mathcal{F}$ since $A^c = \mathcal{F}/A$
Then $A^{c} \cap \{T \le t\} = \mathcal{F} \cap \{T \le t\} – A \cap \{T \le t\}$
Since $\mathcal{F} \cap \{T \le t\} = \mathcal{F}_t$
Then $A^{c} \cap \{T \le t\} = \mathcal{F}_t – A \cap \{T \le t\}$
ie $A^{c} \in \mathcal{F}_T$
However for this to hold we must consider the situation where $A \cap \{T \le t\} = \mathcal{F}_t$
In other words here $A^c$ would be $\emptyset$ and this is where I'm stuck.
If I could prove $\emptyset \in \mathcal{F}_T$ (2) would follow so any hints to get me started would be really appreciated as I really don't know where to start, or if I am completely wrong that would also be good to know! Thanks in advance!
Best Answer
\begin{multline} 1) \emptyset : \emptyset \cap \{T\leq n\} =\emptyset \in \mathcal{F}_n \Rightarrow \emptyset \in \mathcal{F}_n \end{multline}
\begin{multline} 2) \Omega : \Omega \cap \{T\leq n\} =\{T\leq n\} \in \mathcal{F}_n \Rightarrow \Omega \in \mathcal{F}_n \end{multline}
\begin{multline} 3) A \in \mathcal{F}_n : A \cap \{T\leq n\} \in \mathcal{F}_n \forall n \end{multline}
We have to prove that $A^c \cap \{T\leq n\}\in \mathcal{F}_n$ \begin{multline} 4) A^c \cap \{T\leq n\} = \underbrace{(A^c \cap \{T\leq n\})^c}_{\in \mathcal{F}_n} \cap \underbrace{\{T\leq n\}}_{\in \mathcal{F}_n}\in \mathcal{F}_n \end{multline}
\begin{multline} 5) A_k \in \mathcal{F}:(\bigcup^{\infty}_{k=1}A_k) \cap \{T\leq n\} = \bigcup^{\infty}_{k=1} \underbrace{(A_k \cap \{T\leq n\})}_{\in \mathcal{F}_n}\in \mathcal{F}_n \end{multline}