Minimizing the second moment of the difference of two dependent Poisson variables

Question

I want to solve this minimization problem $$\underset{\tilde{a}_j}{\operatorname{min}}E \mid \mid \mathcal{P}_{j}(a_{j}) - \tilde{\mathcal{P}}_{j}(\tilde{a}_{j}) \mid \mid^{2},$$ where $\mathcal{P}_{j}$ and $\tilde{\mathcal{P}}_{j}$ are dependent Poisson random variables. If they are independent, it is easy to prove that it is solved by taking $\tilde{a}_j= -\frac{1}{2}+a_j$ but I did not find any clue how to solve it when they are dependent because of the covariance. Could anyone give a hint please!!

Answer 1

If $\mathcal{P}(t)$ is a Poisson process then for example $\tilde{\mathcal{P}}(t):=\mathcal{P}(at)$ where $a>0$ is a Poisson process that is correlated to the original with a paramter $a$. But that is just one option, you need to be more specific about the form of dependence the two processes have.