Question about the intensity of a cox process, Diggle–Moraga–Rowlingson–Taylor (2013)

Question

On page 2 of "Spatial and spatio-temporal log-Gaussian Cox processes: Extending the geostatistical paradigm" by Diggle–Moraga–Rowlingson–Taylor (2013), accessible at arXiv, they claim the following on the bottom right of the page:

"...in the stationary case the intensity of the Cox process is equal to the expectation of Λ(x)"

My assumption is that by "intensity", they are referring to some extension of $$ \lambda (t)=\lim _{h\downarrow 0}{\frac {1}{h}}\mathbb {E} [N(t+h)-N(t)|{\mathcal {F}}_{t}],$$ as taken from wikipedia at https://en.wikipedia.org/wiki/Intensity_of_counting_processes.

I mentioned "extension" because in the Diggle–Moraga–Rowlingson–Taylor paper, they define the Cox process on $\mathbb {R}^2$, whereas the intensity definition from Wikipedia is to do with processes defined on $\mathbb{R}$.

My issue is that I'm unable to find a proof of their claim. If someone more experienced in this field could explain to me why the authors' claim is true / guide me to a relevant source, I would really appreciate it. Thank you.

Answer 1

The intensity function of a point process $(N_B)$ over $\mathbb R^d$ is defined as the density of the intensity measure $\mu$ of $(N_B)$ relative to the Lebesgue measure over $\mathbb R^d$ (in your case, the Lebesgue measure over $\mathbb R^2$).

In turn, the intensity measure $\mu$ of $(N_B)$ is defined by the formula $\mu(B):=EN_B$ for all Borel sets $B$, in your case all Borel subsets $B$ of $\mathbb R^2$.

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Now, conditionally on a nonnegative-valued process $\Lambda=(\Lambda_x)_{x\in\mathbb R^2}$, the Cox process $(N_B)$ over $\mathbb R^2$ is an inhomogeneous Poisson process with intensity $(\Lambda_x)$. So, for all Borel subsets $B$ of $\mathbb R^2$
$$E(N_B|\Lambda)=\int_B\Lambda_x\,dx,$$ whence $$EN_B=EE(N_B|\Lambda)=E\int_B \Lambda_x\,dx=\int_B E\Lambda_x\,dx.$$ Thus, $(E\Lambda_x)_{x\in\mathbb R^2}$ is indeed the intensity function of the Cox process; the stationarity assumption is not needed here.