Martingale - monotone class theorem
I want to show by using a monotone class argument that the following equality (i.e. it is a martingale) holds for all bounded and -measurable random variables :
, . (*)
where is an Brownian motion (and it follows from (*) also a Brownian motion).
is the enlargement of the filtration by the filtration generated by a r.v. . So far I have shown that
(*) holds for where is bounded and measurable; and is bounded and -measurable. Now, I think I need a monotone class argument in order to show that (*) holds for all bounded and -measurbale random variables.
But unfortunately I am not very familiar in using monotone class arguments.
How do I find the sets needed in the monotone class theorem?
Thanks in advance.