Suppose we have a biased simple random walk $S_K=X_1+\dots+X_k$ on integers which begins at $0$, where $X_l$s are i.d.d random variables such that $P(X=1)=p=1-P(X=-1)>1/2$. It's easy to check $EX=p-q$, $EX^2=4pq$
The one sided stopping time is defined to be $T=\inf \{k : S_k=b\}$ for $b>0$. From the law of large number, one can show $P(T<\infty)=0$.
To solve for $ET^2$, I find this martingale $$W_n=(S_n-n(p-q))^2-n(4pq)$$
And now I want to apply optional stopping theorem. However, none of the criterions in the wiki page of optional stopping theorem is trivially satisfied, since $S_k$, $T$ and $W_n$ are unbounded in sampling spaces.
I can find the answer using generating functions, but I wonder if I could process using this martingale. So far I can tell $EW_{T\land n}=EW_0$, but not $EW_T=EW_0$. I wonder If there is any way to conclude $ET^2<\infty$ without complicated computation, which is enough to derive the answer.