Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $T>0$
- $I:=(0,T]$
- $(\mathcal F_t)_{t\in\overline I}$ be a filtration on $(\Omega,\mathcal A)$
- $U$ and $\tilde U$ be infinite-dimensional separable $\mathbb R$-Hilbert spaces
- $M$ and $N$ be $U$- and $\tilde U$-valued square integrable continuous $\mathcal F$-martingales on $(\Omega,\mathcal A,\operatorname P)$, respectively
- $(e_n)_{n\in\mathbb N}$ and $(\tilde e_n)_{n\in\mathbb N}$ be orthonormal bases of $U$ and $\tilde U$, respectively
We can show that $$\left[\langle M,e_m\rangle_U,\langle N,e_n\rangle_{\tilde U}\right]_t=\int_{(0,\:t]}\frac{{\rm d}\left[\langle M,e_m\rangle_U,\langle N,e_n\rangle_{\tilde U}\right]}{{\rm d}([M]+[N])}\:{\rm d}([M]+[N])\tag1$$ for all $t\in\overline I$ almost surely for all $(m,n)\in\mathbb N^2$, where $[\;\cdot\;]$ and $[\;\cdot\;,\;\cdot\;]$ denote the quadratic variation and covariation, respectively.
I want to show that $$\sum_{(m,\:n)\in\mathbb N^2}\frac{{\rm d}\left[\langle M,e_m\rangle_U,\langle N,e_n\rangle_{\tilde U}\right]}{{\rm d}([M]+[N])}e_m\otimes e_n\tag1$$ exists in the space $\mathfrak L_1(U,\tilde U)$ of nuclear operators from $U$ to $\tilde U$, where $u\otimes\tilde u:=\langle\;\cdot\;,u\rangle_U\tilde u$ for $(u,\tilde u)\in U\times\tilde U$.
By definition of the nuclear norm, this would be the case if $$\sum_{(m,\:n)\in\mathbb N^2}\left|\frac{{\rm d}\left[\langle M,e_m\rangle_U,\langle N,e_n\rangle_{\tilde U}\right]}{{\rm d}([M]+[N])}\right|\tag2$$ exists.
The claim that $(1)$ exists can be found in Theorem 8.2 of Stochastic Partial Differential Equations with Lévy Noise by Peszat and Zabczyk. However, I don't see why the claim is true.