I try to obtain the cumulated intensity measure condition presented in the work from Schmidt, 2009, "Catastrophe Insurance Modeled by Shot-Noise processes" by applying Campbell's Formula.
I have the following idea: we start with Campbell’s formula for point processes, which gives: $$ \mathbb{E} \left[ \int f(x) \, N(dx) \right] = \int f(x) \, \mu(dx). $$ Here, $ \mu(dx) $ is the mean measure.
I want to obtain: $$ \mathbb{E} \left[ \int_0^\infty Y_s \, dN_s \right] = \mathbb{E} \left[ \int_0^\infty Y_s \, \mathcal{L}(ds) \right]. $$ for a predictable process Y. This defines $\mathcal{L}(ds)$ as the $\textit{cumulated intensity measure}$ in the Cox process setting, with $ \mathcal{L}_t = \int_0^t \mathcal{L}(ds)$ as the $\textit{cumulated intensity process}$.
Idea: We rewrite the expectation as: $$ \mathbb{E} \left[ \int_0^\infty Y_s \, dN_s \right] = \mathbb{E} \left[ \mathbb{E} \left( \int_0^\infty Y_s \, dN_s \, \Big| \, \mathcal{L} \right) \right]. $$ Conditioned on $\mathcal{L}$, $ \ N_t$ is a Poisson process with intensity $$\mathcal{L}(ds) $$. We apply Campbell’s formula in this conditional setting: $$ \mathbb{E} \left( \int_0^\infty Y_s \, dN_s \, \Big| \, \mathcal{L} \right) = \int_0^\infty Y_s \, \mathcal{L}(ds). $$
Can you please tell me whether this is correct? I feel like I am missing something.
