Proving Martingale for exponential Random walk

Question

For any $u$, $e^{uS_n - nh(u)}$ where $h(u)=\ln{\mathbb{E}(e^{u\xi _i})}$. Given that $ S_n = S_0 + \sum ^N _{i=1} \xi _i$.

Required to show the proof.

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I need to show

$\mathbb{E}(m_{n+1}|f_n)=m_n$

Working $m_{n+1}$:

$ = \frac{e^{uS_n +1}}{\mathbb{E}[u \xi_i]}$

Then I don't know how to simplify.

Answer 1

$$E[m_{n+1} \mid \mathcal{F}_n] = E[e^{uS_n + u \xi_{n+1} - (n+1) \ln E e^{u \xi_{n+1}}} \mid \mathcal{F}_n] = e^{uS_n} E[e^{-u \xi_{n+1}}]^{-n} = m_n.$$