Stochastic integral with respect to a predictable process again predictable?

Question

Suppose $M$ is a local martingale and $\alpha$ is an $\mathbb{R}$-valued predictable process. Then we know that $\langle M\rangle$ is predictable, but is the stochastic integral $\alpha\bullet \langle M\rangle$ again predictable?

Answer 1

Yes, $\alpha\bullet\langle M\rangle$ is predictable, because $$ \alpha\bullet\langle M\rangle =(\alpha\bullet\langle M\rangle)_- + \Delta(\alpha\bullet\langle M\rangle) =(\alpha\bullet\langle M\rangle)_- +\alpha\cdot\Delta\langle M\rangle, $$ and the above last two terms are both predictable.