If $S,T$ are stopping times in discrete-time, then I know $S+T$ is a stopping time. Is it necessary that $S-T$ is a stopping time? Intuitively, I can see that $S-T$ involves some events happening in the future, and hence not a stopping time. Can anyone give me some examples?
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Your intuition is correct, $S-T$ is not necessarily a stopping time. For a counterexample, consider the simple random walk $(X_n)$ on $\mathbb Z$ started at zero, and let $S=\inf\{n:X_n=1\}$ and $T=1$. Note that $\{S-T\le1\}=\{S\le2\}$ which is not $X_1$-measurable.
Jason
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Why ${S-T\le1}={S\le2}$ is not $X_{1}$-measurable? – Diego Fonseca Nov 30 '19 at 06:01
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Uh, in this particular case, it turns out it is $X_1$-measurable, oops. If we instead assume $(X_n)$ is a lazy simple random walk (so the jump sizes are $-1,0,1$ with probability $1/3$ each), then ${S\le2}={X_1=1}\cup{X_1=0,X_2-X_1=1}$, which is not $X_1$-measurable since $X_2-X_1$ is independent of $X_1$. – Jason Dec 05 '19 at 22:10