Hi !

Looks like it's the same as last year : I can't wait for the next class to solve a problem ^^'

We've studied martingales for only 3 hours already, so I'm still very new to it.

So we have a sequence $\displaystyle (Y_n)_{n\geq 1}$ of iid rv's such that $\displaystyle P(Y_n=-1)=q$ and $\displaystyle P(Y_n=1)=p$

where $\displaystyle p+q=1 \ , \ 0<p<q<1$

Let $\displaystyle X_0=0 \ , \ Z_0=1$... $\displaystyle X_n=Y_1+\dots+Y_n$ and $\displaystyle Z_n=\left(\tfrac qp\right)^{X_n}$

Let's consider the filtration $\displaystyle \mathcal{F}_n=\sigma(Z_0,\dots,Z_n)$

I have to show that $\displaystyle Z_n$ is a positive martingale.

3 points to show :

1/ $\displaystyle Z_n$ is $\displaystyle \mathcal{F}_n$-measurable

2/ $\displaystyle Z_n$ is integrable

3/ $\displaystyle E(Z_{n+1}\mid \mathcal{F}_n)=Z_n$

Point 1/ is okay.

Point 2/ is okay.

Point 3/ is totally unknown Which method to use ?

Thanks for any help/hints