Suppose you have a random walk $S_n = \sum_{i=1}^n X_i$ where $$X_i = \begin{cases} 1 &\hbox{ with probability 1/3}\\ 0 &\hbox{ with probability 1/6} \\ -1 &\hbox{ with probability 1/2} \end{cases}$$ We also set $S_0 := 0$. Let $Z = \sup_n S_n$. Prove that $Z < \infty$ a.s. Use the martingale $M_n = \exp(\theta S_n - n\psi(\theta))$ to compute $P(Z= k)$ for $k=0,1,...$ what is the distribution of $Z$?
I think I need to use the optional stopping theorem but I am not really sure how to proceed. Can someone provide some advice?