predictable stopping time
A predictable, or previsible stopping time is a random time which is possible to predict just before the event. Letting be a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable space , then, a stopping time is predictable if there exists an increasing sequence of stopping times satisfying the following.
The sequence is said to announce or foretell .
For example, if is a continuous adapted process with , such as Brownian motion, then the first time at which it hits a given level is a predictable stopping time. In this case, if is the first time at which hits the level , then the sequence announces .
On the other hand, if is a Poisson process then the first time at which it is nonzero is not predictable. To show this, suppose that are stopping times. The fact that is a martingale means that Doob’s optional sampling theorem can be applied, giving . Then, for gives . So, with probability one, and the sequence cannot announce .
This can be generalized to an arbitrary index set , where a stopping time is predictable if there exists an increasing sequence of stopping times such that whenever is not equal to a minimal element of , and contains no elements of .
|Title||predictable stopping time|
|Date of creation||2013-03-22 18:37:19|
|Last modified on||2013-03-22 18:37:19|
|Last modified by||gel (22282)|
|Synonym||previsible stopping time|