 Solution

Inductively, $X_0,\dots ,X_n$ are functions of ${\Delta }_0,\dots ,{\Delta }_n$, so ${\mathcal{F}}_n\subseteq \sigma ({\Delta }_0,\dots ,{\Delta }_n)$. The average $f_n(X_0,\dots ,X_n)$ is therefore ${\mathcal{F}}_n$-measurable and independent of ${\Delta }_{n+1}$. If $X_0,\dots ,X_n$ are integrable, thenStarting with $X_0={\Delta }_0$, induction proves integrability of every $X_n$. Adaptation holds by definition of the natural filtration. Independence and zero mean now legitimately giveHence the process is a martingale. Only first moments are used; no second-moment or boundedness assumption on the increments is needed.