Martingale - monotone class theorem

Juju

New member
Joined
Feb 5, 2011
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4
Hello,

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 \(\displaystyle \mathcal{G}_s\)-measurable random variables \(\displaystyle Y_s\):
\(\displaystyle E[Y_s(W_t-W_s)]=0\), \(\displaystyle s\leq t\). (*)
where \(\displaystyle W\) is an \(\displaystyle \mathcal{F}_s\) Brownian motion (and it follows from (*) also a \(\displaystyle \mathcal{G}_s\) Brownian motion).

\(\displaystyle \mathcal{G}_s\) is the enlargement of the filtration \(\displaystyle \mathcal{F}_s\) by the filtration generated by a r.v. \(\displaystyle X\). So far I have shown that
(*) holds for \(\displaystyle Y_s=f(X)H_s\) where \(\displaystyle f\) is bounded and measurable; and \(\displaystyle H_s\) is bounded and \(\displaystyle \mathcal{F}_s\)-measurable. Now, I think I need a monotone class argument in order to show that (*) holds for all bounded and \(\displaystyle \mathcal{G}_s\)-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.
 
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