Derivative of the quadratic variation of Levy process

Question

Let $L(t)$ be a n-dimensional Levy process having the decomposition $$ L(t) = \int_{B} x \widetilde{N}(t,dx) $$ where $B=\{ |x|<1 \}$ and $\widetilde{N}(dt,dx) = N(dt,dx) - \nu(dx)dt$ is the compensated Poisson random measure of $N$. In Applebaum's "Lévy Processes and Stochastic Calculus" the quadratic variation of $L(t)$, after some rewriting, is given as $$ \langle L \rangle_t = \int_0^t \int_B xx^T N(ds,dx) $$ So far so good. I'm interested in studying the derivative of the quadratic variation $ Q_t = \frac{d}{dt} \langle L \rangle_t $. In Peszat and Zabczyk's "Stochastic Partial Differential Equations with Levy Noise" I found that it should be that $Q_t$ does not depend neither on time (this I can see, somehow) nor on $\omega$. As far as I can see, $\langle L \rangle_t$ depends on $\omega$ since it contains a random measure, so its derivative should also depend on $\omega$. Moreover, the authors affirm that it should be that $Q_t = Q/\text{Tr}(Q)$, where $Q$ is the covariance operator of $L$ and I don't see how this can be possible. How can I reconcile these two conflicting statements?