0$, and let $\mathcal H_t=(t_k|t_k\leq t )$ the subset of event times between time $0$ and time $t$. Let $N(t)=\sum_{k\geq 1}1_{t_k\leq t}$ denote the number of events between time $0$ and time $t$, where $1_{A}=1$ if $A$ is true, and 0 otherwise. Assume that $N(t)$ is a counting process with conditional intensity function $\lambda(t)$, that is for any $t\geq 0$ and any infinitesimal interval $dt$ $$\Pr(N(t+dt)-N(t)=1|\mathcal H_t)=\lambda(t)dt.\label{eq:counting}$$ Consider another counting process $\tilde{N}(t)$ with the corresponding $(\tilde{t}_k)_{k\geq 1}, \mathcal{\tilde{H}}_t, \tilde{\lambda}(t)$. Then, $N(t),\tilde{N}(t)$ are mutually-exciting Hawkes processes  [@self_exc_HP] if the conditional intensity functions $\lambda(t)$ and $\tilde{\lambda}(t)$ take the form $$\begin{aligned}
\lambda(t)=\mu + \int_0^t g_\phi(t-u)\, d\tilde{N}(u)\label{eq:hawkesintensity}\qquad
\tilde{\lambda}(t)=\tilde{\mu} + \int_0^t {g}_{\tilde{\phi}}(t-u)\, d{N}(u)\end{aligned}$$ where $\mu=\lambda(0)>0, \tilde{\mu}=\tilde{\lambda}(0)>0$ are the base intensities and $g_\phi,g_{\tilde{\phi}}$ non-negative kernels parameterised by $\phi$ and $\tilde{\phi}$. This defines a pair of processes in which the current rate of events of each process depends on the occurrence of past events of the opposite process.

Assume that $\mu=\tilde{\mu},\, \phi= \tilde{\phi}$ and $g_\phi(t) \geq 0$ for $t > 0$, $g_\phi(t)=0$ for $t<0$. If $g_\phi$ admits a form of fast decay then this results in strong local effects. However, if it prescribes a peak away from the origin then longer term effects are likely to occur. 