0
$\begingroup$

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.

  • 0
    Could you provide a little more context here? The simplest form of a martingale stops when you run out of money (ie, your variable becomes 0) and has only a single stopping time. – 2017-01-19
  • 0
    @barrycarter Maybe my question arises more from a theoretical point of view but I am not sure. I was perusing the properties of stopping times and came across this [link](http://www.math.uah.edu/stat/prob/Stop.html), specifically point number 16. I do not quite understand how τ1 and τ2, both, are stopping times. Is there a practical explanation to it or is it just for mathematical purposes. – 2017-01-19
  • 1
    @Insignificant Stopping times are just random variables. $\tau_1$ and $\tau_2$ are two random variables that also happen (i.e. are defined) to be stopping times. – 2017-01-20

1 Answers 1

2

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.

  • 2
    You can also answer the second part of his question using your examples, i.e. a new stopping time defined as the minimum/maximum of "fortune exceeds $100" and "play 17 rounds" is also a stopping time. – 2017-01-20
  • 1
    Thanks @Olorun! Makes so much sense to me now. – 2017-01-20