In Limit theorems for stochastic processes, by Jacod and Shiryaev, they state 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.

There is a similar result for general local martinagles. However, I'm faced with the following problem in my script and I'm not sure why this is true: Suppose we have a continuous local martingale $M$, hence it is of course a locally square-integrable martingale. Moreover $N$ is a general local martingale.

Are we still allowed to write $\langle M,N\rangle $?

this is done in my script, but I do not see why this is allowed. So can we define a predictable quadratic covariation for a general local martingale and a locally square integrable martingale?

I add another example, where this is used: 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 $d\langle Z, M\rangle_s$ ."

So it seem to be true, but could someone give me a reference for this result.


In case you're still interested in this, here's an answer:

In general, $\langle M,N\rangle$ is the dual predictable projection (equivalently, the compensator obtained from the Doob-Meyer decomposition of a locally integrable finite variation process) of $[M,N]$. The dual predictable projection is described in chapter six of volume two of Rogers & Williams: Diffusions, Markov processes and martingales.

The dual predictable projection $B$ of a finite variation process $A$ is well-defined whenever $A$ is locally integrable, and is uniquely determined by having $A-B$ a local martingale. Thus, the predictable covariation $\langle M,N\rangle$ is well-defined whenever $[M,N]$ is locally integrable. This is the case whenever $M$ and $N$ are both locally square-integrable, but may also occur in other situations.

In the case you describe, $M$ is a continuous local martingale and $N$ is a local martingale. As $\Delta[M,N] =\Delta M\Delta N = 0$, $[M,N]$ is continuous. Therefore, it is also locally integrable, and thus the dual predictable projection $\langle M, N\rangle$ is well-defined.


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