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I'm studying Stochast integration and stochastic differential equation by Protter and got confused by the solution of the very first exercise provided here.

So

Exercise 1. Let $S,T$ be stopping times, $S \leq T$ a.s. Show $\mathcal{F}_S \subset \mathcal{F}_T$

The provided solution starts like this:

$\forall A \subset \mathcal{F}_S, \forall t \geq 0, A \cap \{T \leq t\} = (A \cap \{S \leq t\}) \cap \{T \leq t\}$ since $\{T \leq t\} \subset \{S \leq t\}$

But $S \leq T$ almost surely. It means that there is a set $N \subset \mathcal{F}_0$ (usual hypotheses) with $\mathbf{P}(N) = 0$ such that $ \forall \omega \in N : S(\omega) > T(\omega)$ so $\exists t_0 : \{T \leq t_0\} \not\subset \{S \leq t_0\}$

Thus the start of the proof need to be corrected like that:

$\forall A \subset \mathcal{F}_S, \forall t \geq 0, A \cap \{T \leq t\} = (A \cap (\{S \leq t\} \cup N)) \cap \{T \leq t\}$ since $\{T \leq t\} \subset \{S \leq t\} \cup N$

The rest of the solution works if we replace $\{S \leq t\}$ by $\{S \leq t\} \cup N$:

Since $A \cap (\{S \leq t\} \cup N) \subset \mathcal{F}_t$ (since $N \subset \mathcal{F}_0 \subset \mathcal{F}_t$) and $\{T \leq t\} \subset \mathcal{F}_t$ we have $A \cap \{T \leq t\} \subset \mathcal{F}_t$ so $\mathcal{F}_S \subset \mathcal{F}_T$

Am I right or it is an unnecessary complication of the correct solution?

zer0hedge
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  • They probably assume early on that $\mathcal F_0$ (hence, and every other sigma-algebra considered) is complete, that is, that it contains every $N$ such that there exists $A$ in $\mathcal F$ such that $P(A)=0$ and $N\subseteq A$. – Did Mar 12 '17 at 14:45
  • Yes, they do say that. Would it be correct then to assume that we can simply ignore all those $\omega \in A$ because they belongs to all $\sigma-$algebras and thus can not impact the proof? – zer0hedge Mar 12 '17 at 15:14
  • Hmmm... those $\omega$s in $A\subseteq\Omega$ certainly do not belong to the sigma-algebras on $\Omega$. – Did Mar 12 '17 at 15:21
  • Ok, they belong to "all $\sigma$-algebras in the given filtration $(F_t)_{0 \leq t \leq \infty}$ since $F_0$ contains all P-null sets. Right? – zer0hedge Mar 12 '17 at 15:26
  • No, $\omega$ in $\Omega$ belongs to no sigma-algebra on $\Omega$. – Did Mar 12 '17 at 15:32
  • Sorry, but I don't understand what you mean. But anyway thank you for pointing to completenece - I believe that's what I missed. – zer0hedge Mar 12 '17 at 15:36
  • The point is basic: a sigma-algebra on $\Omega$ is made of subsets of $\Omega$, not of elements of $\Omega$. – Did Mar 12 '17 at 15:42
  • Actually, the fact that probability space is complete (you meant probability space, not sigma-algebra, above, right?) does not help. ${T \leq t} \not \subset {S \leq t}$ and you have to extend the solution with $N$ as described above. – zer0hedge Mar 19 '17 at 17:28

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