The random time used is typically a stopping time.
If the process has left limits (http://planetmath.org/CadlagProcess) for every , then it can alternatively be stopped just before the time , resulting in the pre-stopped process
Stopping is often used to enforce boundedness or integrability constraints on a process. For example, if is a Brownian motion and is the first time at which hits some given positive value, then the stopped process will be a continuous and bounded martingale. It can be shown that many properties of stochastic processes, such as the martingale property, are stable under stopping at any stopping time . On the other hand, a pre-stopped martingale need not be a martingale.
For continuous processes, stopping and pre-stopping are equivalent procedures. If is the first time at which , for any given real number , then the pre-stopped process will be uniformly bounded. However, for some noncontinuous processes it is not possible to find a stopping time making into a uniformly bounded process. For example, this is the case for any Levy process (http://planetmath.org/LevyProcess) with unbounded jump distribution.
Stopping is used to generalize properties of stochastic processes to obtain the related localized property. See, for example, local martingales.
|Date of creation||2013-03-22 18:37:38|
|Last modified on||2013-03-22 18:37:38|
|Last modified by||gel (22282)|