In this paper here, the authors defined the probability-generating functional for a counting process $N_t$ as \begin{align*} G[u(t)] = \mathbb{E}\left[\exp\left\{\int \log u(t) dN_t\right\}\right] \end{align*} Specifically for a time inhomogeneous Poisson process with rate $\lambda(t)$, this simplifies to \begin{align*} G[u(t)] = \exp\left\{\int (u(t) - 1)\lambda(t) dt\right\} \end{align*} The authors say that the PGF exhibits properties similar to the moment generating function less the existence of an inverse transformation, but one could still compute a series expansion of the probability density function. The authors do not expand on how to do so, so my question is could anyone help explain the steps in computing the series expansion?
In addition, and this is the step that's more relevant to my research problem, the authors also state the joint PGF between two inhomogeneous Poisson processes with joint variation as \begin{align*} G(u_1, u_2) = \exp\left\{\int(u_1(s)-1)\lambda_1(s) ds + \int(u_2(t)-1)\lambda_2(t) dt + \iint (u_1(s)u_2(t)-1)\lambda_c(t,s) dsdt\right\} \end{align*} I would like to know the series expansion for the joint distribution from this as well. I'm guessing this second problem would be much harder, since this involves some series reversion in two variables, which is especially tricky. Any help appreciated!
