Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 34 / 4 / b / Solution

Here $X_1\le 1$, so for every $\tau >0$,The Jensen inequality for the strictly convex exponential function givesIn fact the inequality is strict: $\mathbb{P}(X_1=1)>0$ and $\mathbb{E}{X}_1=0$ exclude a constant random variable. In particular, $\boxed{1\le M(\tau )<\infty }$ as required.

For $Z_n=e^{\tau S_n}/M(\tau {)}^n$, independence and identical distribution giveThus $Z_n$ is positive, integrable with mean one, and adapted to the natural filtration. Conditioning on the next increment givesConsequently $\boxed{(Z_n)\text{ is a martingale}.}$ This is the exponential martingale of a random walk.