The question is stated as follows:
Let's say $\xi_n$ - i.i.d. random values, $P(\xi_n = 1) = p, P(\xi_n = 0) = 1 - p$; $S_n$ is random walk generated by them, i.e. $S_0 = 0, S_1 = \xi_1, S_n = \sum_{i=1}^n \xi_i$; let $X_n$ be a process defined as $X_n = e^{aS_n + bn}, (a, b) \in \mathbb{R}^2$. Find all pairs $(a, b)$ such that a) $X_n$ is a martingale, b) $X_n$ is a submartingale.
I have an idea of what martingale is, and teacher said that it's a very simple problem, yet I was struggling with it, I feel like maybe I'm using the notion of conditional mean in the wrong way. Here is my take:
We need $\mathbb{E}(X_n | X_1, \ldots , X_{n-1}) = X_{n-1} \Longrightarrow \mathbb{E}(e^{a(\xi_1 + \ldots + \xi_{n-1})+a\xi_n - bn}) = \xi_{n-1} \Longrightarrow \\ e^{a(\xi_1 + \ldots + \xi_{n-1})}e^{ap - bn} = \xi_{n-1} \Longrightarrow \\ a(\xi_1 + \ldots + \xi_{n-1}) + ap - bn = \ln\xi_{n-1}$
What's next? It leads to equality of real numbers $a, b$ and random values, so it feels wrong. Thank you.
HINT: let $\mathcal{F}^{X}_n:=\sigma (X_0,\ldots ,X_n)$ the natural filtration of $\{X_n\}_{n\in\mathbb{N}}$ and note that
$$ \mathbb{E}[X_{n+1}|\mathcal{F}^X_n]=\mathbb{E}[X_ne^{a\xi _{n+1}+b}|\mathcal{F}^{X}_n]=X_n\mathbb{E}[e^{a\xi _{n+1}+b}|\mathcal{F}^{X}_n]\\[1em] \therefore\quad \mathbb{E}[X_{n+1}|\mathcal{F}^X_n]=X_n \iff \mathbb{E}[e^{a \xi _{n+1}+b}|\mathcal{F}^{X}_n]=e^{b}\mathbb{E}[e^{a \xi _{n+1}}|\mathcal{F}^{X}_n]=1 $$
Now note that, as $\mathcal{F}^{X}_n\subset \sigma (\xi _1,\ldots ,\xi _n)$, then $\xi _{n+1}$ is independent of $\mathcal{F}^{X}_n$, so the above reduces to $\mathbb{E}[e^{a \xi _{n+1}}]=e^{-b}$.