Here is the urn model:
At time zero there are $r$ red and $g$ green balls in an urn. At each time-step, we draw out a ball at random and replace it along with $c$ of the same color and $d$ of the opposite color. If $d\neq 0$ is this a supermartingale or a submartingale?
This is my attempt. Let $G_n$ be the number of green balls at time $n$ and $R_n$ the number of red balls at time $n$. Then $G_n+R_n = r+g +n(c+d).$ Let $M_n=\frac{G_n}{R_n+G_n}=\frac{G_n}{r+g+n(c+d)}$, be the portion of green balls in the urn at time $n$. Then I have:
$$ E(M_n|\mathcal{F}_n)=\frac{G_n+c}{r+g+(n+1)(c+d)}\frac{G_n}{r+g+n(c+d)} + \frac{G_n+d}{r+g+(n+1)(c+d)}\frac{R_n}{r+g+n(c+d)}. $$
Where $\mathcal{F}_n$ is just the filtration for $M_n$.
Here is where I'm stuck. I'm not sure how to show that this is greater than or less than $M_n$.
Any help would be appreciated.