Firstly, I just want to check my understanding:
If we have a symmetric random walk such that
$P(S_{n+1} = S_n + 1|S_n)=1/2$
$P(S_{n+1} = S_n - 1|S_n)= 1/2$
then $S_n$ is a martingale, $(S_n)^2$ is a sub-martingale and because a martingale has a constant expectation we subtract the mean $(n)$ to get a martingale. So, $(S_n)^2 - n$ is a martingale. Is that correct?
Secondly, I have a question about an asymmetric random walk:
If we have a random walk such that
$P(S_{n+1} = S_n + 1|S_n)= p(<1/2)$
$P(S_{n+1} = S_n - 1|S_n)= 1-p = q$
then is $S_n$ a sub-martingale or super-martingale? I am thinking super-martingale because its expectation decreases with time, e.g.
$E[S_1] = E[X_1] = p-q$
$E[S_2] = E[X_1 + X_2] = 2(p-q)$
...
$E[S_n] = E[X_1 + X_2 + ... + X_n] = n(p-q)$
where $(p-q)$ is negative so the expectation is decreasing and
$P(X_n = 1)= p(<1/2)$
$P(X_n = -1)= 1-p = q$
Therefore, I would think we have to add the mean, $n(p-q)$, to get a martingale. So, $S_n + n(p-q)$ would be a martingale. However, my lecturer says that $S_n - n(p-q)$ is a martingale and gives no explanation.
Where have I gone wrong in my thinking?