Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 30K / a / Solution

The process is adapted because $X_0$ is ${\mathcal{F}}_0$-measurable, $A_k$ is ${\mathcal{F}}_{k-1}$-measurable, and $M_k-M_{k-1}$ is ${\mathcal{F}}_k$-measurable. Using the predictability of $A_n$ and the martingale increment property,The assumed integrability therefore proves that $X$ is the martingale transform of $M$ by $A$ and is a martingale.