Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 4 / a / Solution

Definefor $t\ge 1$, with $A_0=M_0=0$. The supermartingale property makes $\Delta A_t\ge 0$, and it is ${\mathcal{F}}_{t-1}$-measurable, so $A$ is previsible and nondecreasing. The increments of $M$ have conditional mean zero, so $M$ is a martingale. Finally,and summation gives the Doob decomposition in discrete time