$\sigma$-field of an increasing sequence of stopping times

Question

Let us consider the sequence $(\tau_n)_{n\in \mathbb{N}}$ of stopping times that takes values in $\mathbb{N}$ such that $\tau_n \uparrow \tau$, and $\tau < \infty$. Prove the following equality:

$\mathcal{F}_\tau=\sigma(\cup_n \mathcal{F_{\tau_n}})$

I'm having problem with both the inclusions, any suggestions?

Answer 1

Hints:

1. Let $S,T$ be two stopping times (with respect to a common filtration). Show: If $S \leq T$, then $\mathcal{F}_S \subseteq \mathcal{F}_T$.
2. Conclude that $$\sigma \left( \bigcup_{n \in \mathbb{N}} \mathcal{F}_{\tau_n} \right) \subseteq \mathcal{F}_{\tau}.$$
3. Prove that $\{\tau_n = \tau\} \in \mathcal{F}_{\tau}$ for all $n \in \mathbb{N}$.
4. Show that for any $F \in \mathcal{F}_{\tau}$ and any $n \in \mathbb{N}$ it holds that $F \cap \{\tau_n=\tau\} \in \mathcal{F}_{\tau_n}$.
5. Conclude that $$F = \bigcup_{n \in \mathbb{N}} (F \cap \{\tau=\tau_n\}) \in \sigma \left( \bigcup_{n \in \mathbb{N}} \mathcal{F}_{\tau_n} \right) $$ for any $F \in \mathcal{F}_{\tau}$.