Theorem

Let $M$ be a right continuous $L^{2}$-martingale. Let $X\in\Lambda^{2}(\mathscr{P},M)$. $\forall t\ge 0$ let $$Y_{t}=\int\limits \mathbb{1}_{[0,t]}X \, dM$$Then $(Y_{t})_{t\ge 0}$ is a $L^{2}$-martingale and hence admits a right continuous version.