Suppose $S_n = (X_n,Y_n)$ is a symmetric random walk on $\mathbb{Z}^2$. Show that $G_n = X_n^2 + Y_n^2 - n$ is a martingale. What is true about $E_{(x_0,y_0)}[|S_n|]$? Find an upper bound for $E_{(0,0)}[T_r]$ using optional stopping, where $T_r$ is the first time that $|S_n| \geq r$.
Here is what I have: $$ \begin{align} G_{n+1} - G_n &= X_{n+1}^2 + Y_{n+1}^2 - (n+1) - (X_n^2 + Y_n^2 - n)\\ &= (X_{n+1}^2 - X_n^2) + (Y_{n+1}^2 - Y_n^2) - n - 1 + n\\ &= (X_{n+1} - X_n)^2 + (Y_{n+1} - Y_n)^2 + 2X_{n+1}X_n + 2Y_{n+1}Y_n - 1 \end{align} $$ We want to show that $E[G_{n+1} - G_n | A_v] = 0$, where $A_v = {S_n,S_{n-1},...S_0}.$ $$ \begin{align} E[G_{n+1} - G_n | A_v] &= E[(X_{n+1} - X_n)^2|X_n] + E[(Y_{n+1} - Y_n)^2|Y_n] + 2E[X_{n+1}X_n|X_n] + 2E[Y_{n+1}Y_n|Y_n] - 1\\ &= 1 + 1 - 1 + ? \end{align} $$ I'm stuck here (is the above correct so far?), and not sure how to proceed with the rest of the problem.
let $\mathcal{F}_n=\sigma(X_1,Y_1,\dots,X_n,Y_n)$, $\xi_{n+1}=X_{n+1}-X_{n}$ and $\eta_{n+1}=Y_{n+1}-Y_n$. Then, $$ G_{n+1}=X_{n}^2+2X_{n}\xi_{n+1}+\xi_{n+1}^2+Y_{n}^2+2Y_{n}\eta_{n+1}+\eta_{n+1}^2-n-1,$$ hence $$G_{n+1}=G_{n}+2X_{n}\xi_{n+1}+2Y_{n}\eta_{n+1}+\xi_{n+1}^2+\eta_{n+1}^2-1. $$ One knows that $G_n$ is $\mathcal{F}_n$-measurable, that $\xi_{n+1}$ and $\eta_{n+1}$ are independent of $\mathcal{F}_n$, that $E[\xi_{n+1}]=E[\eta_{n+1}]=0$, and that $E[\xi_{n+1}^2]+E[\eta_{n+1}^2]=1$, hence $$E[G_{n+1}|\mathcal{F}_n]=G_n+2X_{n}E[\xi_{n+1}]+2Y_{n}E[\eta_{n+1}]+E[\xi_{n+1}^2]+E[\eta_{n+1}^2]-1=G_n. $$ Thus, $(G_n)$ is a martingale with respect to the filtration $(\mathcal{F}_n)$.
By the way, I guess that the random walk you mean is simple random walk because the result is not true for every random walk, just check that the expectation of $G_{n+1}$ is not equal to $G_{n}$ in general.