c$)$ Let $({}_k{)}_k=1,2,$cdots,$N$ be a $(F_k{)}_k=0,1,$cdots,$N*$ predictable process and define the discrete$-$time process $,(M_n{)}_n=1,2,$cdots,$N$ by $M_n={\displaystyle \underset{k=1}{\overset{n}{}}}{}_k(x_k-x_{k-1}).$ Prove that $(M_n{)}_n=1,2,$cdots,$N$ is discrete$-$time martingale with respect to the filtration $(F_n{)}_n=0,1,$cdots,$N$

Solution:

This is Theorem $3\mathrm{.}11.$ in different notations.