 Solution

Because $H_k$ is previsible, $H_k$ is ${\mathcal{F}}_{k-1}$-measurable. The finite sum ${\hat{X}}_n$ is therefore ${\mathcal{F}}_n$-measurable. Boundedness of $H$ and integrability of the martingale increments implyFinally,Thusso $\hat{X}$ is a martingale. It is the martingale transform of $X$ by $H$.