Let $0 < p < 1$. Let $\{X_n\}_{n\in\mathbb N}$ be a sequence of independent random variables such that
$$P (X_n = 1) = p, P (X_n = −1) = 1 − p$$
for all $n \in \mathbb N$. For each $n\in\mathbb N$, set
$$S_n := X_1 + X_2 + · · · + X_n$$
for the filtration $F_n := \sigma(X_i,1 \leqslant i \leqslant n).$
Now I define the following stochastic process $\{M_n\}$ by
$$M_n := (\frac{1-p}{p})^{S_n},\forall n \in\mathbb N,$$
I want to show that $M_n$ is an $F_n$-martingale.
I have tried to prove the following,
$$E=[M_{n+1}|F_n]=M_n. $$
However, by inserting $M_n$, I don't get the desired result.
Best Answer
$E(\frac {1-p} p)^{S_{n+1}}|\mathcal F_n)=(\frac {1-p} p)^{S_n}E(\frac {1-p} p)^{X_{n+1}}=(\frac {1-p} p)^{S_n} [(\frac {1-p} p) p+\frac p {1-p} (1-p)]=(\frac {1-p} p)^{S_n}$