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If $\{X_n\}_{n\in\mathbb N_0}$ is a Markov chain is $T:=\{\inf n\ge1:X_{n}=X_{n-1}\}$ a stopping time ?

$\{T=n\}=\{X_0\neq X_1, X_1\neq X_2\dots X_{n-2}\neq X_{n-1},X_{n-1}=X_n\}$, I would say yes, and what if I change the indices in the set by one, i.e. $\tilde T:=\{\inf n\ge1:X_{n}=X_{n+1}\}$, then information up to time $n+1$ is necessary, but is not $T-1=\tilde T$ ?

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  • $\begingroup$ What filtration do you consider, is it the filtration generated by process $X_n$? $X_n$ is an arbitrary process or do you have any assumption about it? $\endgroup$
    – M. Stawiski
    Oct 16, 2015 at 11:10
  • $\begingroup$ @ M. Stawiski $\mathcal F_n=\sigma(X_0,\dots,X_n)$ $\endgroup$
    – ketum
    Oct 16, 2015 at 11:12
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    $\begingroup$ Yes $T$ is a stopping time for the natural filtration of $(X_n)$ and $\tilde T=T-1$ is not. Is this your question? $\endgroup$
    – Did
    Oct 16, 2015 at 11:27
  • $\begingroup$ @Did Yes but $\tilde T$ can be also determined by $T$, why it is not ? $\endgroup$
    – ketum
    Oct 16, 2015 at 11:30
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    $\begingroup$ $\widetilde{T}$ is in fact $T-1$ but that doesn't mean than it is a stopping time. Real life intuition example: consider $T$ as a moment when you stop driving a car. You can define $T$ as a moment when your car crash. To avoid crashing you can consider $\widetilde{T} =T-1$ as a moment when you get out of car just a moment (hour, minute) before you car crashes but it's not a stopping time because you don't know for sure that you will crash in 1 hour (assuming that you won't do it for purpose). $\endgroup$
    – M. Stawiski
    Oct 16, 2015 at 11:32

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