# Variance of geometric random walk

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 have 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 $n$?

• I am not following your math at all. Why are you multiplying step sizes? – dsaxton Nov 14 '16 at 3:05
• Hint (possibly): consider the log of the walker's position. That way, you're just adding instead of multiplying, which means you will have a normal distribution. – barrycarter Nov 14 '16 at 3:12