# show $\sup\{t \in \mathbb{N}_0 : S_t =1\}$ is a stopping time

Let $$(X_n)_{n \in \mathbb{N}_0}$$ be a sequence of independent and identically distributed random variables with $$\mathbb{P}(X_1 = 1) = \mathbb{P}(X_1 = -1) = \frac{1}{2}.$$ Define $$S_t = \sum\nolimits_{k=1}^t X_k$$ for $$t\in \mathbb{N}$$ and $$S_0 = 0$$

Show that $$\sigma := \sup\{t \in \mathbb{N}_0 : S_t =1\}$$ and $$\tau := \inf\{t \in \mathbb{N}_0 : S_t =1\}$$ are stopping times with respect to the natural filtration of $$(X_n)$$

I need to show that $$\{\sigma = t\} \in F_t$$ where $$F_t$$ is $$\sigma(X_0, ... , X_t)$$

I wanted to write $$\{\sigma = t\}$$ differently, but I don't see how this can be done.

Thanks for any help.

• Are you sure that this is what you have to prove? I doubt that $\sigma$ is a stopping time. – saz Oct 30 '18 at 11:59
• Hmm.. is it because $\sigma=\infty$ a.s.? I.e., for each $t<\infty$, $\{\sigma\le t\}$ is null, so it's trivially measurable w.r.t. $\mathcal F_t$? – AddSup Oct 30 '18 at 14:43

1. $$\tau$$ is a stopping time: At time $$t$$ it is known if $$\{\tau=t\}$$ occurred. $$\{\tau=t\}=\{S_0\neq 1,\dots,S_{t-1}\neq 1,S_t=1\}.$$
2. $$\sigma$$ is not a stopping time: At time $$t$$ you can not know if $$\{\sigma=t\}$$ occurred. $$\{\sigma=t\}=\{S_t=1,S_{t+1}\neq 1,S_{t+2}\neq 1\dots\}.$$