Past exam of the probability, measure and martingales course of the University of Oxford / 2023 / 2 / c / i

Each $Z_n$ is a bounded Borel measurable function of $X_n$, including the specified value at $X_n=1$. Therefore $(Z_n)$ is an adapted process for $({\mathcal{F}}_n)$ and consists of integrable random variables.

Fix an atom of a sigma-algebra in ${\mathcal{F}}_n$ with positive probability, corresponding to the dyadic interval $[u,u+\delta )$, with $\delta =2^{-n}$. By the uniform distribution, each half has conditional probability $1/2$. The conditional expectation of $Z_{n+1}$ on that atom is thereforeThe middle values cancel. On the zero-probability atom the conditional expectation can be chosen as zero, agreeing with our version of $Z_n$. Henceso $(Z_n)$ is a martingale. This is the dyadic slope martingale associated with the Lipschitz function $f$.