Stopping Time
Does a "stopping time" refer to a type of event? The time that this event occurs is random, and it is a stopping time if, at any point in time, you know whether the event has occurred or not. At any time, I know whether I am ruined or not. For instance, I am not ruined right now.
Hitting times like the second example above can be important examples of stopping times. The reason being that at a given time you don't know where the process will go next. You would not be able to completely stop the universe this way, because as of right now, zero degrees Kelvin absolute zero has not been reached, and it may not be possible. Another conceivable way is to freeze the universe, thus slowing molecular motion. It lets everything exist, and change, and I doubt imagining something outside of it is easy for time-bound beings like us. Conversely, suppose that this condition holds. This is also true:.
I don't know when ruin occurs, or if it will occur at all, but if it does, I will know. Suppose I am driving along a very long road, and that I'm looking for the parking spot which is furthest towards the other end of the road call this "the last parking spot". I pass by available spots along the way, but at any time, I never know if I have passed the last free parking spot. I could just have passed some empty spot, but I cannot see if there are more empty spots later on, and I wouldn't know if the spot that I just passed was the last one or not.
This is a deterministic stopping time. At any time, I know whether or not my birthday has occurred this year.
In fact, I know exactly when my birthday occurs, which makes this a non-typical stopping time in the sense that it is deterministic. The reason being that at a given time you don't know where the process will go next. You seem to understand the concept pretty well. But in this precise example, the time of ruin is the first time that you have exactly 0.
The concept of stopping time is closely related to that of filtration of a stochastic process. By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service , privacy policy and cookie policy , and that your continued use of the website is subject to these policies. Home Questions Tags Users Unanswered. What is meant by a stopping time?
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I am having difficulty understanding what a stopping time is. We are then given an example: Time of ruin is a stopping time. I don't quite understand what this is supposed to tell us.
Stopping time
If this is fine so far, then I have issues with the example. Aaron Hall 6 A stopping time is not an event, it is a random variable. For a filtration, the following definition is appropriate. Here's a review of how this is done: Filtrations can also be completed. The last definition must seem awfully obscure, but it does have a place. In the theory of Markov processes , we usually allow arbitrary initial distributions, which in turn produces a large collection of distributions on the sample space. In light of the previous result, the next definition is natural.
Right continuous filtrations have some nice properties, as we will see later.
If the original filtration is not right continuous, the slightly refined filtration is:. The reason for this is to preserve the meaning of time converging to infinity. In a sense, a stopping time is a random time that does not require that we see into the future. The term stopping time comes from gambling. Consider a gambler betting on games of chance. The gambler's decision to stop gambling at some point in time and accept his fortune must define a stopping time. That is, the gambler can base his decision to stop gambling on all of the information that he has at that point in time, but not on what will happen in the future.
The terms Markov time and optional time are sometimes used instead of stopping time. This is very simple. So, the finer the filtration, the larger the collection of stopping times. But this filtration corresponds to having complete information from the beginning of time, which of course is usually not sensible. The converse to part a or equivalently b is not true, but in continuous time there is a connection to the right-continuous refinement of the filtration. The following corollary now follows.
The converse to part c of the result above holds in discrete time.
The Annals of Probability
Conversely, suppose that this condition holds. The following theorems give some basic ways of constructing new stopping times from ones we already have.
This is a corollary of the previous theorem. Here is the appropriate definition:.