(Almost) complete characterization of stability of a discrete-time Hawkes process with inhibition and memory of length two

We consider a discrete-time version of a Hawkes process defined as a Poisson auto-regressive process whose parameters depend on the past of the trajectory. We allow these parameters to take on negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n{)}_{n\ge 0}$ with parameters $a_1,\dots ,a_p\in \mathbb{R}$, $p\in \mathbb{N}$ and $\lambda \ge 0$, such that for all $n\ge p$, conditioned on $X_0,\dots ,X_{n-1}$, $X_n$ is Poisson distributed with parameter $${(a_1{X}_{n-1}+\cdots +a_p{X}_{n-p}+\lambda )}_{+}$$ We consider specifically the case $p=2$, for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the linear recurrence equation $x_n=a_1{x}_{n-1}+a_2{x}_{n-1}+\lambda$ remains bounded, but the converse is not true. Relatedly, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$, in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.