Variance of geometric random walk

Question

I am trying to calculate the mean and variance of the following simple random walk: suppose we start from 1. With probability $p$ it can increase to $a$, and with probability $q(=1-p)$ decrease to $b$. The walk follows these steps, therefore in the next step it can three outcomes: $a^2$ with probability $p^2$, $ab$ with probability $2pq$, and $b^2$ with probability $q^2$. I managed to calculate the mean of step $n$: $E_n=(ap+bq)^n$. What is the variance of step $n$?

Answer 1

Let $X$ be the number of steps with the multiplicative factor $a$ in $n$ steps. Then $$X \sim \text{Binomial}(n, p)$$ The position after $n$ steps can be expressed as $$ Y = a^Xb^{n-X}$$ Therefore

$$ E[Y^m] = \sum_{x=0}^n a^{mx}b^{m(n-x)}\binom {n} {x} p^x q^{n-x} = \sum_{x=0}^n \binom {n} {x} (a^mp)^x (b^mq)^{n-x} = (a^mp + b^mq)^n $$

And thus $$ Var[Y] = E[Y^2] - E[Y]^2 = (a^2p + b^2q)^n - (ap + bq)^{2n}$$