# Questions about proving the statement: $\tau(\omega)$ is a stopping time, iff $\{ \tau(\omega) < t\}$ $\in \mathcal{F}_t$, for all $t \geq 0$.

Assume the filtration is right-continuous ($$\mathcal{F}_{t+0} := \cap_{s>t}\mathcal{F_s} = \mathcal{F}_t$$) and complete, then we have that $$\tau(\omega)$$ is a stopping time, if and only if $$\{ \tau(\omega) < t\}$$ $$\in \mathcal{F}_t$$, for all $$t \geq 0$$.

Here is the proof:

$$\Rightarrow$$: $$\{ \tau(\omega) < t\}$$ = $$\cup_{n=1}^{\infty} \{ \tau(\omega)\leq t- 1/n\} \in \mathcal{F}_t;$$

The "$$\in$$" is truth due to the definition of stopping time and the definition of sigma algebra; that is $$\{ \tau(\omega)\leq t- 1/n\} \in \mathcal{F}_{t-1/n} \subset \mathcal{F}_t$$ for all $$n$$, then the union is in $$\mathcal{F_t}$$.

$$\Leftarrow$$: $$\{ \tau(\omega) \leq t\}$$ = $$\cap_{n=1}^{\infty} \{ \tau(\omega) < t+ 1/n\} \in \mathcal{F}_{t+0} = \mathcal{F}_t;$$

One question is that:

The last equality follows from our right continuous assumption, but the why the "$$\in$$" is the case?

Another question is that:

Many books give the statement $$\cap_{s>t}\mathcal{F_s} = \cap_{n=1}^{\infty}\mathcal{F}_{t+1/n}$$, but they don't provide any proof of it, I've tried to prove it, but it doesn't make sense to me. can you give any thought about it?

## 1 Answer

Equivalently, $$\bigcap_{n=1}^{\infty} \{ \tau(\omega) < t+ 1/n\} \in \mathcal{F}_s$$ for every $$s > t$$, which is implied by, since $$\{ \tau(\omega) < t+ 1/n\}$$ is a decreasing sequence of events, that $$\{ \tau(\omega) < t+ 1/n\} \in \mathcal{F}_s$$ for all $$n > N$$ for some $$N$$. To see this just set $$N = 1/(s-t)$$.

For the second question, $$\bigcap_{s>t}\mathcal{F_s} \subseteq \bigcap_{n=1}^{\infty}\mathcal{F}_{t+1/n}$$ follows immediately from the fact that $$t+1/n > t$$ so the left intersection is over more sets.

The other direction is the archimedean property. Fix $$s > t$$. Then there is some positive integer $$m$$ such that $$t+1/m < s$$ showing $$\bigcap_{n=1}^{\infty}\mathcal{F}_{t+1/n} \subseteq \mathcal{F}_{t+1/m} \subseteq \mathcal{F}_s.$$ Now just intersect over all $$s>t$$.

• i've been thinking this question for a long time, you save me, thanks! Commented Jul 9, 2020 at 3:53