Let $X_1, X_2,\ldots$ i.i.d. random variables such that $X_i \in \{-1,0,1,\ldots\}$ with $\mathbb{P}[X_1 = -1]>0$ and $S_n = X_1+\cdots + X_n$.
Let $m(\lambda) = \mathbb{E}[e^{-\lambda X_1}]$ and define
$$M_n = e^{-\lambda S_n}\cdot m(\lambda)^{-n}$$
and
$$f(s) = \min\{\lambda > 0: \ m(\lambda)^{-n} < s\}.$$
- Show that $M_n$ is martingale.
- Prove that, if $m(\lambda) >1$, the martingale $M_n$ is bounded until the time $$T_k = \min \{n\in \mathbb{N}:\ S_n = -k\}.$$
- Use the optional stopping theorem to prove $$\mathbb{E}[s^{T_k}] = e^{-kf(s)}.$$
I've already prove that $M_n$ is martingale. I'm having trouble trying to prove the second and third point, I would apprecciate any hint I can recieve to begin both proofs because I'm totally lost.
I assume $T_k$ is stopping time, and if I understood right, second point asks to show $\mathbb{E}[\vert M_{T_k}\vert] \leq \mathbb{E}[\vert M_n\vert]$ for any $T_k < n$, am I right?