Consider a process with an initial state $(A_0, B_0)$ and at each step $t+1$, considering $(A_t, B_t)$:
- $(A_{t+1}, B_{t+1}) = (A_t + 1, B_t)$, with probability $A_t/(A_t + B_t)$
- $(A_{t+1}, B_{t+1}) = (A_t, B_t + 1)$, otherwise (with probability $B_t/(A_t + B_t)$)
Using a martingale, what is the expected value of $A_t$ after $t$ steps as a function of $A_0$, $B_0$, and $t$?
Is it a good approach to envision a martingale to which we could apply the optional stopping theorem?
EDIT: We can write a recursion as (given that $A_t + B_t = A_0 + B_0 + t$):
$$E[A_{t+1}\mid A_t] = A_t\left(1 + \frac{1}{A_0 + B_0 + t}\right)$$
Which can be solved to yield (considering $E[A_0] = A_0$):
$$E[A_t] = A_0\left(1 + t\frac{X_0}{A_0 + B_0}\right)$$
The question is what function $f$: $Z_t = f(A_t)$ can be such that $Z_t$ is a martingale with respect to $A_t$?