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Hi I got some problem with the proof of this lemma which is left as excercise Let $(F_t)$ be a filtration and S,T be stopping times, then show $ \{S=T\},\{S≤T\},\{S<T\} \in F_S\bigcap F_T$. Could any one show to how to do it ? thanks in advance.

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up vote 2 down vote accepted

By symmetry, you just have to check that these events all belong to $\mathcal{F}_T$.

Let's do it for $\{T = S\}$ : by definition of $\mathcal{F}_T$, you have to prove that $\{T = S \} \cap \{T \le n \}$ belongs to $\mathcal{F}_n$ for all $n \ge 0$.

But $\{T = S \} \cap \{T \le n \} = \bigcup_{k \le n} \{T =k \} \cap \{S = k\}$. As both $\{T = k\}$ and $\{S =k \}$ belong to $\mathcal{F}_k \subset \mathcal{F}_n$, we have the result.

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