Let $U,H$ be a separable $\mathbb R$-Hilbert spaces, $M$ be a $U$-valued square-integrable martingale on a filitered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ with martingale covariance $Q$, $\Phi:\Omega\times[0,T]\to\mathfrak L(U,H)$ be a predictable process with $$\operatorname E\left[\int_0^T\left\|\Phi_sQ_s^{1/2}\right\|_{\operatorname{HS}(U,\:H)}^2\:{\rm d}[M]_s\right]<\infty$$ and $\Phi\cdot M$ denote the Itō integral process.

Above $[M]$ denotes the (scalar) quadratic variation of $M$, i.e. the unique nondecreasing adapted process such at $\left\|M\right\|_H^2-[M]$ is a square-integrable martingale. Moreover, if $[\![M]\!]$ denotes the tensor-quadratic variation of $M$, i.e. the unique adapted process with values in the space of nuclear, nonnegative and self-adjoint operators on $U$ such that $M\otimes M-[\![M]\!]$ is a square-integrable martingale, then $$[\![M]\!]_t=\int_0^tQ_s\:{\rm d}[M]_s\;\;\;\text{for all }t\in[0,T]\;.$$

Now, if $N$ is another $U$-valued square-integrable martingale, can we find an expression for the covariation $[\Phi\cdot M,N]$?

I'm also interested in an expression for the tensor-covariation $[\![\Phi\cdot M,N]\!]$ where $N$ might take values in another separable $\mathbb R$-Hilbert space $\tilde H$.

All I was able to find in books are the classical results $$[\Phi\cdot M,\Psi\cdot M]_t=\int_0^t\langle\Phi_sQ_s^{1/2},\Phi_sQ_s^{1/2}\rangle_{\operatorname{HS}(U,\:H)}\:{\rm d}[M]_s$$ and $$[\![\Phi\cdot M]\!]_t=\int_0^t\Phi_sQ_s\Phi_s^\ast\:{\rm d}[M]_s\;.$$

Remark: Cross-posted on mathematics.


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