Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 32 / 2 / a / Solution

The moment-generating function satisfies $0<\varphi (\lambda )<\infty$. The proposed process is nonnegative and adapted to the natural filtration of the random walk, and independence of the increments givesMoreover $X_{n+1}$ is independent of ${\mathcal{F}}_n$, soTherefore $M^{\lambda }$ is a mean-one nonnegative martingale, the exponential martingale of a random walk. For $\lambda =0$ it is the constant process one.