Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / A / 2 / a / ii

Write ${\Delta }_0=X_0$ and ${\Delta }_j=X_j-X_{j-1}$ for $j\ge 1$. If $i<j$, then ${\Delta }_i$ is ${\mathcal{F}}_{j-1}$-measurable. The martingale property and linearity of conditional expectation giveThe product ${\Delta }_i{\Delta }_j$ is integrable by Cauchy-Schwarz inequality. The pull-out property of conditional expectation, followed by its tower property, therefore givesFor unbounded ${\Delta }_i$, the pull-out identity follows by truncation and Cauchy-Schwarz inequality. This proves martingale-difference orthogonality, including orthogonality to the initial value.

Expand $X_n={\sum }_{j=0}^n{\Delta }_j$ in the Hilbert space $L^2$. Orthogonality gives the exact identityThe terms are nonnegative and $\mathbb{E}{X}_0^2<\infty$, so boundedness of the left side over $n$ is equivalent to convergence of the series on the right.