Consider stopping times $S$ and $T$ for a filtration $(\mathcal{F}_n)$. Show that $\mathcal{F}_{\min(S,T)} = \mathcal{F}_{S}\cap \mathcal{F}_{T}$.

Question

I'm trying to solve this question but my argument works for $\mathcal{F}_{\max(S,T)} = \mathcal{F}_{S}\cap \mathcal{F}_{T}$. I'm wondering if anyone can confirm if this question is a typo and should be asking for $\max(S,T)$ instead of $\min(S,T)$.

Answer 1

It holds that

$$\mathcal{F}_{\max\{S,T\}} = \sigma(\mathcal{F}_S,\mathcal{F}_T),$$

see this answer for a proof, and

$$\mathcal{F}_{\min\{S,T\}} = \mathcal{F}_S \cap \mathcal{F}_T$$

as is shown e.g. in this answer.

SO: No, there is no typo in the question. Note that, in general,

$$\mathcal{F}_{S} \cap \mathcal{F}_T \subsetneq \sigma(\mathcal{F}_S,\mathcal{F}_T).$$