Stopping Time - Examples

Examples

To illustrate some examples of random times that are stopping rules and some that are not, consider a gambler playing roulette with a typical house edge, starting with $100:

• Playing one, and only one, game corresponds to the stopping time = 1, and is a stopping rule.
• Playing until he either runs out of money or has played 500 games is a stopping rule.
• Playing until he is the maximum amount ahead he will ever be is not a stopping rule and does not provide a stopping time, as it requires information about the future as well as the present and past.
• Playing until he doubles his money (borrowing if necessary if he goes into debt) is not a stopping rule, as there is a positive probability that he will never double his money. (Here it is assumed that there are limits that prevent the employment of a martingale system, or a variant thereof, such as each bet being triple the size of the last. Such limits could include betting limits but not limits to borrowing.)
• Playing until he either doubles his money or runs out of money is a stopping rule, even though there is potentially no limit to the number of games he plays, since the probability that he stops in a finite time is 1.

Hitting times can be important examples of stopping times. However, while it is relatively straightforward to show that essentially all stopping times are hitting times (Fischer, 2013), it can be much more difficult to show that a certain hitting time is a stopping time. The latter types of results are known as the Début theorem.