I'm working on my master thesis now and I've been struggling with a problem for some while now and no one seems to be able to help me or point me in any direction. So now I reach out to see if someone here can help me.
Basically I have two (dependent) compound Poisson processes with no parameters in common. I want to find an analytical expression of the covariance of those processes or at least a useful approximation. What remains to compute is this the following:
Searched: $E[Y^a Y^b]$, where $Y^a = \sum_{i=1}^{N^a} X_i^a$ and $Y^b = \sum_{i=1}^{N^b} X_i^b$
Known distributions: $N^a \sim Po(\lambda^a)$, $N^b \sim Po(\lambda^b)$, $X^a \sim Exp(1/\mu^a)$, $X^b \sim Exp(1/\mu^b)$
Known parameters: $\lambda^a, \lambda^b, \mu^a, \mu^b$
Correlations: $Cov(N^a,N^b)$ and $Cov(X^a,X^b)$ are non-zero and can be found. Otherwise independent ($N^a \perp X^a$, etc).
How can I solve this? Does any analytically tractable solution exist? Can I approximate the solution in any way?
If anything is unclear please let me know and I'll explain further!
Thanks in advance!