# Martingale - monotone class theorem

• Jun 28th 2013, 12:10 PM
Juju
Martingale - monotone class theorem
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 $\mathcal{G}_s$-measurable random variables $Y_s$:
$E[Y_s(W_t-W_s)]=0$, $s\leq t$. (*)
where $W$ is an $\mathcal{F}_s$ Brownian motion (and it follows from (*) also a $\mathcal{G}_s$ Brownian motion).

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