-algebra at a stopping time
Let be a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable space . For every , the -algebra represents the collection of events which are observable up until time . This concept can be generalized to any stopping time .
The reason for sampling at time rather than at is to include the possibility that , in which case is not defined.
This can be shown as follows. If is a progressively measurable process, then the stopped process is also progressive. In particular, is -measurable and is -measurable. Conversely, if is -measurable then is a progressive process and is -measurable. By letting increase to infinity, it follows that is -measurable for every -measurable random variable . Now suppose also that is adapted, and hence progressive. Then, is -measurable. Letting increase to infinity shows that is -measurable.
As a set is -measurable if and only if is an -measurable random variable, this gives the following alternative definition,
From this, it is not difficult to show that the following properties are satisfied
Any stopping time is -measurable.
If for all then .
If are stopping times and then . In particular, if then .
If are stopping times and then .
if the filtration is right-continuous and are stopping times with then . More generally, if eventually then this is true irrespective of whether the filtration is right-continuous.
If are stopping times with eventually then . That is,
In continuous-time, for any stopping time the -algebra is the set of events observable up until time with respect to the right-continuous filtration . That is,
If are stopping times with whenever is not a maximal element of , and then,
The -algebra of events observable up until just before time is denoted by and is generated by sampling predictable processes
Suppose that the index set has minimal element . As the predictable -algebra is generated by sets of the form for and , and for , the definition above can be rewritten as,
Clearly, . Furthermore, for any stopping times then when restricted to the set .
If is a sequence of stopping times announcing (http://planetmath.org/PredictableStoppingTime) , so that is predictable, then
|Title||-algebra at a stopping time|
|Date of creation||2013-03-22 18:38:56|
|Last modified on||2013-03-22 18:38:56|
|Last modified by||gel (22282)|