Let $A, B$ be stopping times of a process $X_0, ... X_n$ where $A$ is a stopping time iff $I_{A = k}$ is a function of $X_0, ... X_k$ (where $I$ denotes an indicator variable).
We want to show that:
- $min(A, B)$ is a stopping time.
In other words, I guess we want to show that $I_{min(A, B) = k}$ is a function of $X_0, ... X_k$ given that both $I_{A = k}$ and $I_{B = k}$ are. This might sound like a dumb question, but I am struggling to express $I_{min(A, B) = k}$ in terms of $I_{A = k}$ and $I_{B = k}$. Since $min(A, B)$ is either $A$ or $B$, it seems that $I_{min(A, B) = k}$ is either $I_{A = k}$ or $I_{B = k}$, both of which are functions of $X_0, ... X_k$. I am confused on where to go from here or if what I just pointed is even relevant. Can we do something in terms of set unions of the process $X_0, ... X_k$?
- $A + B$ is a stopping time. Again, we want to show that $I_{A + B = k}$ is a function of $X_0, ... X_k$. I am having the same difficulty as above.
(Please no measure-theoretic results or proofs as I am not familiar with measure theory! I am supposed to work with this definition only.)
$A$ and $B$ are stopping times, so $I_{A=k}$ and $I_{B=k}$ are both determined by $X_0,...X_k$ (this is simply the definition). For the first part it would be convenient to use that also $I_{A>k}$ is solely determined by $X_0,...X_k$: $$I_{\min\{A,B\}>k}=I_{A>k}I_{B>k},$$ So the minimum of the two is termined by $X_0,...X_k$ and hence it is a stopping time.
The second part is just a matter of summing over all the possibilities: $$I_{A+B=k}=\sum_{n=0}^k I_{A=n}I_{B=k-n}.$$
You should be able to conclude the argumentation now.