Martingale problem Theorom 5.10
For part a, I know I'd need to prove that Sn is a martingale. I could do so given the fact that EXi = 0 and E[Sn] = nExi which also equals 0. So E[Sn] = 0. Then I'd show E[Sn/Sn-1] = Sn, so Sn is a martingale. But I don't know how I'd use theorem 5.10 to show the question in part a. I also don't know how I'd show part b either.
Assuming that $a$ is positive, the stopping time $T_a$ is a.s. finite. I let you prove that this is right.
By using the theorem 5.10 with the martingale $(S^{2}_{T_{a}\land n}-T_{a}\land n\cdot\sigma^2)_n$ you can have access to this equation: $$E[S^{2}_{T_{a}\land n}] = E[T_{a}\land n\cdot\sigma^2]$$
Then, tend $n \rightarrow \infty$ with the propper convergence theorems (not the same one for the two expectations) to resolve question a).