We consider here an exponential kernel $$g_\phi(t-u)= \eta e^{-\delta (t-u)}, t>u
\label{expo_kernel}$$ where $\phi=(\eta,\delta)$. $\eta\geq 0$ determines the sizes of the self-excited jumps and $\delta>0$ is the constant rate of exponential decay. The stationarity condition for the processes is $\eta < \delta$. Figure \[fig:process\_and\_inten\] gives an illustration of two mutually-exciting Hawkes processes with exponential kernel and their conditional intensities.

Compound completely random measures
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A homogeneous completely random measure (CRM) [@Kingman1967; @Kingman1993] on $\mathbb R_+$ without fixed atoms nor deterministic component takes the form $$\begin{aligned}
W=\sum_{i\geq 1}w_i \delta_{\theta_i}\end{aligned}$$ where $(w_i,\theta_i)_{i\geq 1}$ are the points of a Poisson process on $(0,\infty)\times\mathbb R_+$ with mean measure $\rho(dw)H(d\theta)$ where $\rho$ is a L' evy measure, $H$ is a locally bounded measure and $\delta_x$ is the dirac delta mass at $x$. The homogeneous CRM is completely characterized by $\rho$ and $H$, and we write $W\sim \operatorname{CRM}(\rho,H)$, or simply $W\sim \operatorname{CRM}(\rho)$ when $H$ is taken to be the Lebesgue measure. @Griffin2017 proposed a multivariate generalisation of CRMs, called compound CRM (CCRM). A compound CRM $(W_1,\ldots,W_p)$ with independent scores is defined as $$\begin{aligned}
W_k=\sum_{i\geq 1}{w_{ik}}\delta_{\theta_i},~~k=1,\ldots,p\end{aligned}$$ where $w_{ik}=\beta_{ik}w_{i0}$ and the scores $\beta_{ik}\geq 0$ are independently distributed from some probability distribution $F_k$ and $W_0=\sum_{i\geq 1}w_{i0}\delta_{\theta_i}$ is a CRM 