Consider a point process $N$. For the linear Hawkes process, the conditional intensity is given by $\lambda(t) = \nu + \int h(t-s) N(ds)$, with constant $\nu > 0$ and kernel $h(s)$. In almost every work I found on the existence of stationary ("stable") solutions (e.g., Hawkes, Daley & Vere-Jones, ...) it is assumed that $h(s)$ is non-negative.
Assuming that we can guarantee that $\lambda(t)$ itself stays non-negative, what stability conditions are known for $h(s)$ (at least partially) negative?
Brémaud and Massoulié (1996) study nonlinear Hawkes processes for which $h(s)$ could take negative values, but then their stability condition (Theorem 1) seems odd because it involves $|h(s)|$. My intuition is that negative parts of $h(s)$ should not prevent a stationary solution. Maybe one needs to make further assumptions about the nonlinearity, e.g. to be monotonically increasing? Again, I couldn't find any relevant literature on this.