A $2D$ random walk starts at $(X_0, Y_0) = (k, k)$ where $k>0$ is an integer. At each step $(X_{n+1}, Y_{n+1}) = (X_{n}-1, Y_{n})$ or $(X_{n+1}, Y_{n+1}) = (X_{n}, Y_{n}-1)$ with the same probability. The process stops when $X_n = 0$ or $Y_n=0$. Let $t$ be the number of moves before stopping. I want to calculate $\Bbb E(t)$. A straight forward calculation gives $$\Bbb E(t) = 2k(1-\frac{C(2k, k)}{2^{2k}}).$$ Apparently there is an elegant solution using martingales.
So far, I've shown that $X_n + \frac{n}{2}$, $Y_n+\frac{n}{2}$ (and their combinations such as $X_n - Y_n$), $(X_n-Y_n)^2 - \frac{n}{2} $, $\frac{C(X_n+Y_n, X_n)}{2^{X_n+Y_n}}$ and $\alpha^{-X_n}\beta^{-Y_n}$ (where $\alpha+ \beta = 2$) are martingales.