Two stopping times : Intuition

Question

I am studying the theory of martingales for the very first time and I am confused about :

1) How can a process have two stopping times?

2) What does is mean(intuitively) when we say, for example that if S and T and two stopping times, then, min(S,T) and max(S,T) are stopping times as well.

Any help is much appreciated!

Thanks.

Answer 1

Think of a stopping time as a rule employed by a gambler for deciding when to quit playing and go home. Which rule the gambler uses may depend on his mood that day. For example, if the gambler's fortune after $n$ turns at the roulette wheel is $X_n$, then one rule might be "play until my fortune exceeds \$100", formalized by the stopping time $S:=\min\{n: X_n>100\}$. Another rule might be "play until I win just once", formalized by $T:=\min\{n\ge 1: X_n-X_{n-1}>0\}$. Another might be "play 17 rounds", formalized by $R:=17$.

To qualify as a stopping time, the decision to stop must be based on what has happened so far. Thus, $U:=\min\{n:X_{n+10}>100\}$ is not a stopping time, because the decision to stop now requires knowledge of what will happen 10 rounds in the future.