I consider $$Z_n(c):= \exp\{c S^*_n- f(-c)n\}$$ with $f(k):=\log E(e^{kS_1})$ , $S_i=z+i-\sum\limits_{j=1}^iY_j$, $(Y_j)_j$ are i.i.d, $Y_j \in \mathbb{N}_0$ and $S^*_i:=-S_i$
$z$ should be a fixed value equal or greater than zero.
Edit: I got: $$E[Z_{t+1}| \mathcal{F_t}]= E[Z_t e^{cY_{t+1}-f(-c)}|\mathcal{F_t} ]= Z_t E[e^{cY_{t+1}-f(-c)}|\mathcal{F_t}]= Z_t E[e^{cY_{t+1}-f(-c)}] \overset{?}{=} Z_t$$
How can I verify the last step?
What do I choose for my filtration: $$\mathcal{F_t}=\sigma(S^*_1,....,S^*_t)$$ or
$$\mathcal{F_t}=\sigma(Z_1,....,Z_t)$$