 Solution

Predictability means $K_s$ is ${\mathcal{F}}_{s-1}$-measurable. If $|K_s|\le L$, each product $K_s(M_s-M_{s-1})$ is integrable, and the finite sum defining $Y_t$ is integrable. Pulling the bounded predictable factor out of the conditional expectation givesTherefore the bounded predictable martingale transform $Y$ is a martingale starting at zero.