# Existence of predictable quadratic covariation for a special pair of local martingales

In Limit theorems for stochastic processes, by Jacod and Shiryaev we have the existence of a predictable quadratic covariation process stated as the following theorem

$\mathbf{Theorem}$ To each pair $(M,N)$ of locally square integrable martingales one associates a predictable process $\langle M,N\rangle \in \mathcal{V}$, unique up to an evanescent set, such that $MN - \langle M,N\rangle$ is a local martingale. Moreover, $$\langle M,N\rangle =\frac{1}{4}(\langle M + N,M + N\rangle - \langle M - N,M - N\rangle )$$ and if $M,N\in\mathcal{H}^2$ then $\langle M,N\rangle\in\mathcal{A}$ and $MN -\langle M,N\rangle \in\mathcal{M}$. Furthermore $\langle M, M\rangle$ is non-decreasing, and it admits a continuous version if and only if $M$ is quasi-left-continuous.

where $\mathcal{V}$ is the space of processes with finite variation, $\mathcal{A}$ the space of processes with integrable variation, $\mathcal{H}^2$ the space of square integrable martingales and $\mathcal{M}$ the space of uniformly integrable martingales.

In this paper by Protter and Shimbo they define on page 269 in (4) the processes $Z_t$ and mention that this need not be continuous. Hence $Z$ is a general local martingale. On the same page after $(7)$ they write: "Since $M$ is continuous there is no issue about the existence of $d\langle Z,M\rangle_s$ ."

Why is the existence of the predictable quadratic covariation process still valid for a pair $(M,N)$ of local martingales, one continuous and the other not?

I asked this question also on MSE but no one has answered it yet (I also started a bounty without success.). So I hope this is the right place to ask this question.