Hawkes processes with infinite mean intensity

The stability condition for Hawkes processes and their nonlinear extensions usually relies on the condition that the mean intensity is a finite constant. It follows that the total endogeneity ratio needs to be strictly smaller than unity. In the present Letter we argue that it is possible to have a total endogeneity ratio greater than unity without rendering the process unstable. In particular, we show that, provided the endogeneity ratio of the linear Hawkes component is smaller than unity, quadratic Hawkes processes are always stationary, although with infinite mean intensity when the total endogenity ratio exceeds 1. This results from a subtle compensation between the inhibiting realizations (mean reversion) and their exciting counterparts (trends).