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.