Limit theorems for Hawkes processes including inhibition

Multivariate Hawkes processes on inhomogeneous random graphs

“…To circumvent this, one has to choose a non-negative and nonlinear function f to preserve the non-negativity of the intensity. A classic choice is to take f (x) = max (0, µ + x) (see for instance [9] for estimation model or [23,14] with h in (1.7) signed and with compact support). One can also introduce inhibition through a signed multiplying factor (that may depend or not on the neuron), see for instance [29,28,57].…”