Let

• $ (\Omega,\mathcal A,\operatorname P)$ be a complete probability space
• $ (\mathcal F_t)_{t\ge 0}$ be a complete and right-continuous filtration of $ \mathcal A$
• $ H$ be a separable $ \mathbb R$ -Hilbert space
• $ (e_n)_{n\in\mathbb N}$ be an orthonormal basis of $ H$
• $ \mathfrak L_1(H)$ denote the space of nuclear operators on $ H$
• $ x\otimes y:=\langle\;\cdot\;,x\rangle_Hy\in\mathfrak L_1(H)$ for $ x,y\in H$

If $ M,N$ are $ H$ -valued continuous local $ \mathcal F$ -martingales on $ (\Omega,\mathcal A,\operatorname P)$ , let

• $ [M,N]$ denote the covariation of $ M$ and $ N$ , i.e. the unique (up to indistinguishability) real-valued continuous $ \mathcal F$ -adapted process of locally bounded variation such that $ \langle M,N\rangle_H-[M,N]$ is a local $ \mathcal F$ -martingale. Moreover, $ [M]:=[M,M]$ .
• $ [\![M,N]\!]$ denote the tensor covariation of $ M$ and $ N$ , i.e. the unique (up to indistinguishability) $ \mathfrak L_1(H)$ -valued continuous $ \mathcal F$ -adapted process of locally bounded variation such that $ M\otimes M-[\![M,N]\!]$ is a local $ \mathcal F$ -martingale

If $ g:[0,\infty)\to E$ is a right-continuous function into a $ \mathbb R$ -Banach space $ E$ of locally bounded variation, let $ {\rm d}g$ denote the unique vector measure on $ \mathcal B([0,\infty))$ with $ $ {\rm d}g((a,b])=g(b)-g(s)\;\;\;\text{for all }b\ge a\ge0\;.$ $

We can show that there is a $ \operatorname P$ -null set $ A$ such that $ {\rm d}[M,N](\omega)$ is absolutely continuous with respect to $ {\rm d}([M](\omega)+[N](\omega))$ and hence $ $ [M,N]_t(\omega)=\int_0^tq_s(\omega)\:{\rm d}([M]_s(\omega)+[N]_s(\omega))\tag1$ $ for all $ \omega\in\Omega\setminus A$ . $ q$ can be extended to a $ \mathcal F$ -predictable process on $ (\Omega,\mathcal A,\operatorname P)$ .

If $ (e_n)_{n\in\mathbb N}$ is an orthonormal basis of $ H$ , then $ $ [M,N]=\sum_{n\in\mathbb N}[\langle M,e_m\rangle_H,\langle N,e_n\rangle_H]\;\;\;\text{almost surely}\tag2$ $ and $ $ [\![M,N]\!]=\sum_{n\in\mathbb N}=\sum_{(m,\:n)\in\mathbb N^2}[\langle M,e_m\rangle_H,\langle N,e_n\rangle_H]e_m\otimes e_n\;\;\;\text{almost surely}\tag3\;.$ $ From this and the result above, we obtain $ $ [\![M]\!]_t=\int_0^tQ^M\:{\rm d}[M]_s\;\;\;\text{for all }t\ge0\text{ almost surely}\tag4$ $ for some $ \mathfrak L_1$ -valued $ \mathcal F$ -predictable process $ Q^M$ on $ (\Omega,\mathcal A,\operatorname P)$ .

This can be found in Stochastic Partial Differential Equaitons with Lévy Noise by Peszat and Zabczyk, Theorem 8.2. Therein $ Q^M$ is called the martingale covariance of $ M$ . According to Stochastic Partial Differential Equations by Chow, Page 163 in the second edition, $ Q^M$ is also called the local covariation operator or the local characteristic operator of $ M$ .

According to Chow, it’s also possible to find a representation like $ (4)$ for $ [\![M,N]\!]$ . However, it’s not clear to me if this representation should be an integral with respect to $ {\rm d}([M]+[N])$ (which is clearly possible by the result $ (1)$ above) or $ {\rm d}[M,N]$ .

In any case, besides the two mentioned references, I’m not able to find any textbook or paper which deals with this “martingale covariance” (or however it is called). I’m especially interested in the question above and the role of this operator in the generalization of the Kunita-Watanabe identity (see [How does the Kunita-Watanabe identity generalize to stochastic integration on Hilbert spaces? other question)).

I’d be happy about any reference.