Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 32 / 1 / a / ii / Solution

Use conditional monotone convergence and the tower property of conditional expectation in the preceding construction:This proves the martingale property. The construction gives $M_n\ge 0$ and ${\sup }_n\mathbb{E}{M}_n\le C$, so $M$ is bounded in $L^1$. Taking $p=n$ in the increasing family also gives $M_n\ge X_n^{+}\ge X_n$. Define $Y_n=M_n-X_n$. Then $Y$ is nonnegative and adapted, andThus $Y$ is a supermartingale, with $\mathbb{E}|Y_n|\le \mathbb{E}{M}_n+\mathbb{E}|X_n|\le 2C$. The requested conclusion isThis positive martingale majorant of an L1-bounded submartingale is not asserted to have uniform integrability; boundedness in $L^1$ alone does not imply that stronger property.