I would like a hint for the following problem:
Consider a biased random walk on the integers with probability $p<1/2$ of moving to the right and probability $1-p$ of moving to the left. Let $S_n$ be the value at time $n$ and assume that $S_0=a$, where $0. Show that $M_n=[(1-p)/p]^{S_{n}}$ is a martingale.
I need to show that $\mathbb{E}[M_{n+1}|S_{0}, \dots S_{n}] = M_{n}$. However $ \begin{align} \mathbb{E}[M_{n+1}|S_{0}, \dots, S_{n}] &= \mathbb{E}[[(1-p)/p]^{S_{n+1}}|S_{0}, \dots, S_{n}] \\ &= \mathbb{E}[[(1-p)/p]^{S_{n+1}}|S_{n}] &&\text{(By Markovity.)} \\ &= [(1-p)/p]^{p(S_{n}+1) + (1-p)(S_{n}-1)} \\ &= [(1-p)/p]^{S_{n}+2p-1}, \end{align} $ which is not what I need. So it seems that either I'm making a mistake or the problem is wrong.
Am I making a mistake?
Is $M_n$ a martingale?