Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 28 / 1 / a

Assume $0<p<1$, so the ratio defining $Z_n$ is meaningful. Extend the random walk by $S_0=0$, the filtration by a trivial ${\mathcal{F}}_0$, and the process by $Z_0=1$. Write $r=q/p$. Each $Z_n=r^{S_n}$ is ${\mathcal{F}}_n$-measurable and integrable, since $|S_n|\le n$. The next increment is independent of ${\mathcal{F}}_n$, soTherefore $\boxed{(Z_n{)}_{n\ge 0}\text{ is a nonnegative martingale},\mathbb{E}{Z}_n=1}$. This is the exponential martingale of a biased simple random walk, with exponent $\log (q/p)$. At $p=q=1/2$, it is the constant process one. The endpoint cases $p=0$ or $p=1$ require separate definitions and are not covered by the printed ratio.