proof of hitting times are stopping times for right-continuous processes
If is a probability measure on and represents the completion (http://planetmath.org/CompleteMeasure) of the -algebra with respect to , then it is enough to show that is an -stopping time. By the universal completeness of it would then follow that
for every and, therefore, that is a stopping time. So, by replacing by if necessary, we may assume without loss of generality that is complete with respect to the probability measure for each .
Let consist of the set of measurable times such that for every and that . Then let be the essential supremum of . That is, is the smallest (up to sets of zero probability) random variable taking values in such that (almost surely) for all .
For any set
Clearly, and, choosing any countable dense subset of , the right-continuity of gives
So, , which implies that with probability one. However, by the right-continuity of , whenever is finite and , so
This shows that and therefore whenever . So, almost surely and giving,
So, is a stopping time.
Finally, suppose that does not have a minimum element. Choosing a sequence in then the above argument shows that
are stopping times so,
|Title||proof of hitting times are stopping times for right-continuous processes|
|Date of creation||2013-03-22 18:39:12|
|Last modified on||2013-03-22 18:39:12|
|Last modified by||gel (22282)|