Doob’s optional sampling theorem
for all in the index set . Doob’s optional sampling theorem says that this equality still holds if the times are replaced by bounded stopping times . In this case, the -algebra is replaced by the collection of events observable at the random time (http://planetmath.org/SigmaAlgebraAtAStoppingTime),
In discrete-time, when the index set is countable, the result is as follows.
Doob’s Optional Sampling Theorem.
This theorem shows, amongst other things, that in the case of a fair casino, where your return is a martingale, betting strategies involving ‘knowing when to quit’ do not enhance your expected return.
In continuous-time, when the index set an interval of the real numbers, then the stopping times can have a continuous distribution and need not be measurable quantities. Then, it is necessary to place conditions on the sample paths of the process . In particular, Doob’s optional sampling theorem holds in continuous-time if is assumed to be right-continuous.
|Title||Doob’s optional sampling theorem|
|Date of creation||2013-03-22 16:43:41|
|Last modified on||2013-03-22 16:43:41|
|Last modified by||skubeedooo (5401)|