Independence of Poisson processes having rates depending on the sample path

Question

I am trying to prove an equality and through my computation I have to compute $Var \left[\sum_{j=1}^{J} \mathcal{Y}_{j} \left( \sum_{k=0}^{N-1} a_j(\mathbf{X}(t_k)) (t_{k+1}-t_k) \right) \right]$ where $\mathcal{Y}_{j}$ are independent unit poisson processes, $\mathbf{X}$ is a sample patha and $a_j$ are scalar functionals of the sample path. Using independence, can we claim that $$Var \left[\sum_{j=1}^{J} \mathcal{Y}_{j} \left( \sum_{k=0}^{N-1} a_j(\mathbf{X}(t_k)) (t_{k+1}-t_k) \right) \right]= \sum_{j=1}^{J} \sum_{k=0}^{N-1} a_j(\mathbf{X}(t_k)) (t_{k+1}-t_k) $$ Or this is not true since the rates depndens on the same sample path?? I have the same issue when dealing with the covariance!!! If it is not true, there is any hint to deal with path dependence.