Assume a Markov chain $\{X_n\}_{n \in \mathbb{N}_0}$ with state space $\mathbb{X}$. Show that all of the following represent stopping times:
a) $T = \inf \{k > 10: X_k = x \}, \, x \in \mathbb{X}.$
b) $T = \inf\{k \geq 5 : X_k = X_2 \}$
c) $S = \inf\{k > T : X_k \in A\}, \, A \subset \mathbb{X}, \, T \,\,\text{stopping time}$
Attempt:
a)
Since every constant time is a stopping time, $N = 10$ is a stopping time for $\{X_n\}_{n \in \mathbb{N}_0}$. Thus, from the strong Markov property, the stochastic process is still a Markov chain after time $10$, with the same properties except the initial state. So
$$
T = \inf \{k > 10: X_k = x \}, \, x \in \mathbb{X}.
$$
is basically the first time $\{Y_n\}_{n \in \mathbb{N}_0}$, with $Y_n = X_{10 + n}$, visits $x$, which is a stopping time. But since $T$ is a stopping time for chain $Y$, it must be for $X$ too.
(Same argument works for c).
b)
(Is the same argument adequate for this one?)
$5$ being a constant time is also a stopping time. Also, since we care about times $\geq$ 5, we already know which state the chain visited at time $2$. Calling this state $x$, we have that $T$ can be expressed as in case a, making it a stopping time.
Any elaboration on the validity of these arguments would be greatly appreciated.
Best Answer
Caveat to everything I'm writing below: I assume that you've seen the formal definition of a stopping time in terms of $\sigma$-algebras/filtrations. It's possible that this is coming from a book or class that does not rigorously discuss stopping times in this manner; if so, my answer does not really apply to your question.
This is a tricky one. The spirit of what you've written is correct, but I'm not convinced by the particular arguments. I will note that your instructor (assuming you have one) may disagree.
The main thing that appears to be missing is some invocation of the definition of stopping times. Your task is to show that for all $n$, $\{T = n\} \in \sigma(X_0, \dots, X_n)$; it seems like you're making your own life a bit harder by skipping this definition (which isn't too hard to apply here) and instead defining some new variables while making some assertions that seem not to be proved.
For instance: in your first solution, you correctly assert that the first visit of $Y_n$ to $x$ is a stopping time -- but why is that true? Note that this claim is more or less equivalent to the original problem; indeed, that equivalence seems to be the crux of your argument. It seems likely to me that the fact you're citing was covered in your class; however, I would think -- and again, your instructor may disagree -- that your job is instead to recreate the argument showing why that cited fact was true and apply the argument to this context. I think a proper argument should actually be shorter and feel less hand-wavy than what you have presented here.