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

Question

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.

Answer 1

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. $$ \{\sigma=t\}=\{S_t=1,S_{t+1}\neq 1,S_{t+2}\neq 1\dots\}.$$