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We have the usual $(\Omega, \mathcal{F}, P)$ stochastic basis. Let $\rho, \tau: \Omega \to T \cup \{+\infty\}$ be stopping times and $\mathcal{F}_{\rho}, \mathcal{F}_{\tau}$ their respective $\sigma$-algebras. Prove that $[\rho \le \tau ] \in \mathcal{F}_{\rho} \cap \mathcal{F}_{\tau}$.

Here is what I tried:

Let's first prove that $[\rho \le \tau ] \in \mathcal{F}_t$: $[\rho \le \tau ] = [\rho \le t] \cap [\tau \le t] \in \mathcal{F}_t \space$. Is this correct?

But now I'm having a hard time proving that it lies in the intersection of the $\sigma$-algebras.

Any hints?

Did
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shimee
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    No, the "equality" $[\varrho \leq \tau] = [\varrho \leq t] \cap [\tau \leq t]$ does not hold. On the right-hand side you lost the information whether $\varrho$ is less or equal than $\tau$ or not. (Consider for example $w$ such that $\tau(w)=1$, $\sigma(w)=2$, $t=2$. Then $w \in [\varrho \leq t] \cap [\tau \leq t]$, $w \notin [\varrho \leq \tau]$.) – saz May 24 '13 at 07:05
  • Yeah I thought so. Any ideas how to go about the proof? – shimee May 24 '13 at 09:10

1 Answers1

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To show that $[\rho\leqslant \tau]$ is in $\mathcal F_\rho$, use the identity $$ [\rho\gt\tau]\cap[\rho\leqslant t]=[\rho\leqslant t]\cap\bigcup_{s\in\mathbb Q,s\lt t}[\rho\gt s]\cap[\tau\leqslant s]. $$ Likewise, $[\rho\leqslant \tau]$ is in $\mathcal F_\tau$ since $$ [\rho\gt\tau]\cap[\tau\leqslant t]=[\tau\leqslant t]\cap\bigcup_{s\in\mathbb Q,s\leqslant t}[\rho\gt s]\cap[\tau\leqslant s]. $$

Did
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  • then $[\rho \le \tau] \in \mathcal{F}{\rho} \cap \mathcal{F}{\tau}$ by the complement property of $\sigma$-algebra, right? – shimee May 24 '13 at 18:46
  • Did, can you please explain these identities? Where do they come from? Why can an uncountable set be replaced by a countable set of rational numbers? Thank you. – Ivan Sep 07 '13 at 16:12
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    @Ivan Each one can be proved by the usual double inclusion approach. One uses twice the fact that for every real numbers $a$ and $b$, $a\lt b$ if and only if there exists some rational number $s$ such that $a\lt s\lt b$. – Did Sep 07 '13 at 16:18