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Let $(X_{n})_{n \geq 0}$ be a sequence of random variables and $\tau,t$ stopping times with respect to the sequence $(X_{n})_{n \geq 0}$

$$\begin{align*}\{\tau+t =n\} = \{\tau+t = n\} \cap \{t \leq n\} &= \bigcup_{k=0}^n \{\tau+t = n\} \cap \{t = k\} \\ &= \bigcup_{k=0}^n \{\tau=n-k\} \cap \{t=k\}. \end{align*}$$

As $\{\tau =n-k\} \in \mathcal{F}_{n-k} \subseteq \mathcal{F}_n$ and $\{t = k\} \in \mathcal{F}_k \subseteq \mathcal{F}_n$ for any $k \leq n$, this implies that $\{\tau+t=n\} \in \mathcal{F}_n$, and so $\tau+t$ is a stopping time.

Now, my question is, let assume that I am considering $\tau-t$. I know that in general, $\tau-t$ is not a stopping time. However, if I were to consider my birthday this year (a stopping time), which is a deterministic stopping time. At any time, I know exactly when my birthday occurs. Also, I know two days before my birthday i.e, $\tau-2$. What kind of a formulated counterexample will show that $\tau-2$ is indeed a stopping time in this setting.

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  • $\begingroup$ I'm not sure I understand what you're asking. Are you asking whether or not $\tau-2$ is a stopping time when $\tau$ is deterministic? $\endgroup$ Nov 12, 2021 at 3:28
  • $\begingroup$ @user6247850 Yes. I am asking whether or not τ−2 is a stopping time in this case, and if so, what is your reasoning? $\endgroup$
    – Smith Pay
    Nov 12, 2021 at 3:49
  • $\begingroup$ To be clear, by "in this case" you mean "$\tau$ is deterministic"? $\endgroup$ Nov 12, 2021 at 4:02
  • $\begingroup$ @user6247850 yes. $\tau$ is deterministic. We need to stop two days before my birthday. $\endgroup$
    – Smith Pay
    Nov 12, 2021 at 4:31

2 Answers 2

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If $\tau$ is deterministic, $\tau-2$ is deterministic. As long as $\tau-2 \ge 0$, it is a stopping time because $\{\tau-2 = n\} = \emptyset$ or $\{\tau-2 = n\} = \Omega$ for all $n$.

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I think the key is that your $\tau$ is deterministic, so $\tau-2$ is deterministic. But if $\tau$ is not deterministic, i.e. $\tau$ is not a constant random variable, we want to know if $\{\tau-2\leq n\}$ happened or not, we need the information in $\mathscr{F}_{n+2}$, since $\{\tau-2\leq n\}\equiv\{\tau\leq n+2\}$.

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