Showing martingale property of a series

Question

I want to show that the following series is a martingale.

$P(X_1=1)=P(X_2=-1)=0.5$ and

$P(X_i=X_{i-1})=p$ and $ P(X_i=-X_{i-1})=1-p$

$S_n=X_1+...+X_{n-1}+\frac{1}{2(1-p)}X_n$

We need to show that:

$E[S_{n+1}|\mathcal{F}_{n}]=S_{n}$

Answer 1

This is how I did it:

$S_{n+1}=X_1+...+\frac{1}{2(1-p)}X_{n}+ \frac{2(1-p)-1}{2(1-p)}X_n+\frac{1}{2(1-p)}X_{n+1}=$

$=S_n+\frac{2(1-p)-1}{2(1-p)}X_n+\frac{1}{2(1-p)}X_{n+1}$

Then the conditional expectation:

$E[S_{n+1}|\mathcal{F}_{n}]= E[S_n+\frac{2(1-p)-1}{2(1-p)}X_n+\frac{1}{2(1-p)}X_{n+1}|\mathcal{F}_n]=$

$=S_n+\frac{2(1-p)-1}{2(1-p)}X_n+\frac{1}{2(1-p)}E[X_{n+1}|\mathcal{F}_n]=$

$=S_n+\frac{2(1-p)-1}{2(1-p)}X_n+\frac{1}{2(1-p)}(2p-1)X_n$

$=S_n+\frac{1-2p}{2(1-p)}X_n-\frac{1-2p}{2(1-p)}X_n=S_n$