Solution
= Solution
Predictability means $K_s$ is $\mathcal F_{s-1}$-measurable. If $|K_s|\leq L$, 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> gives
$$
\mathbb E[Y_t-Y_{t-1}\mid\mathcal F_{t-1}]
=K_t\mathbb E[M_t-M_{t-1}\mid\mathcal F_{t-1}]=0.
$$
Therefore \b[the bounded <predictable> <martingale transform> $Y$ is a <martingale> starting at zero].