Example of event not in the $\sigma$-algebra of a stopping time.

Question

I am trying to gain some intuition about the $\sigma$-algebra of a stopping time. Let $\tau$ be a stopping time on a filtration $(F_t)$. Then $F_\tau = \{A \in F : \{\tau \leq t \} \cap A \in F_t \forall t \}$. Can somebody give an example of a stopping time and an event which is not in $F_\tau$?

Answer 1

Let $(S_{n})$ be a simple random walk with $S_{n} = X_{1}+\cdots + X_{n}$ and define the filtration $\mathcal{F}$ by $\mathcal{F}_{n} = \sigma(X_{1},\cdots,X_{n})$.

Define $T$ to be the first hitting time of $1$, that is $\inf\{n\mid S_{n}=1\}$ and $H = \{S_{100} = 0\}$. Then $H$ is not in $\mathcal{F}_{T}$.

To see this, $H \cap \{T \leq 10\}$ contains all paths of the random walk that visits state $1$ at or before time $10$ and visits $0$ at time $100$, so it cannot be determined by $X_{1},\cdots,X_{10}$. Therefore, $H \cap \{T \leq 10\} \notin \mathcal{F}_{10}$ and this implies $H \notin \mathcal{F}_{T}$.